$\sum_{\alpha\in{\mathbb{F}}_{n}^{+}}a_{\alpha}W_{\alpha}W_{\alpha}^{*}$
converges in the operator norm topology.

We denote by $C^{*}({\bf W})$ the $C^{*}$-algebra generated by $W_{1},\ldots
W_{n}$ and the identity. We also consider the lexicographic order on the
unital free semigroup ${\mathbb{F}}_{n}^{+}$.

######  Theorem 5.4.

Let $g=1+\sum_{|\alpha|\geq 1}b_{\alpha}Z_{\alpha}$ be a free holomorphic
function in a neighborhood of the origin such that $b_{\alpha}>0$ and
$\sup_{\alpha\in{\mathbb{F}}_{n}^{+}}\frac{b_{\alpha}}{b_{g_{i}\alpha}}<\infty$
for every $i\in\\{1,\ldots,n\\},$ and let ${\bf W}=(W_{1},\ldots,W_{n})$ be
the weighted left creation operators associated with $g$. Then the following
statements hold.

  1. (i)

The $C^{*}$-algebra $C^{*}({\bf W})$ is irreducible.

  2. (ii)

The $n$-tuples $(W_{1}\otimes I_{\mathcal{H}},\ldots,W_{n}\otimes
I_{\mathcal{H}})$ and $(W_{1}\otimes
I_{{\mathcal{H}}^{\prime}},\ldots,W_{n}\otimes I_{{\mathcal{H}}^{\prime}})$
are unitarily equivalent if and only if
$\dim{\mathcal{H}}=\dim{\mathcal{H}}^{\prime}$.

  3. (iii)

If there is $i\in\\{1,\ldots,n\\}$ such that

$\lim_{\gamma\to\infty}\left(\frac{b_{g_{p}\gamma}}{b_{g_{i}g_{p}\gamma}}-\frac{b_{\gamma}}{b_{g_{p}\gamma}}\right)=0\quad\text{for
any}\quad p\in\\{1,\ldots,n\\},$

then the $C^{*}$-algebra $C^{*}({\bf W})$ contains all the compact operators
in $B(F^{2}(H_{n}))$.

###### Proof.

Let $A\in B(F^{2}(H_{n}))$ be commuting with each operator in $C^{*}({\bf
W})$. Since $A(1)\in F^{2}(H_{n})$, we have
$A(1)=\sum_{\beta\in{\mathbb{F}}_{n}^{+}}c_{\beta}\frac{1}{\sqrt{b_{\beta}}}e_{\beta}$
for some coefficients
$\\{c_{\beta}\\}_{{\mathbb{F}}_{n}^{+}}\subset{\mathbb{C}}$ with
$\sum_{\beta\in{\mathbb{F}}_{n}^{+}}|c_{\beta}|^{2}\frac{1}{b_{\beta}}<\infty$.
On the other hand, since $AW_{i}=W_{i}A$ for any $i\in\\{1,\ldots,n\\}$,
relation (1.7) implies

$\begin{split}Ae_{\alpha}&=\sqrt{b_{\alpha}}AW_{\alpha}(1)=\sqrt{b_{\alpha}}W_{\alpha}A(1)=\sum_{\beta\in{\mathbb{F}}_{n}^{+}}c_{\beta}\frac{\sqrt{b_{\alpha}}}{\sqrt{b_{\alpha\beta}}}e_{\alpha\beta}.\end{split}$

Similarly, since $A^{*}W_{i}=W_{i}A^{*}$ for any $i\in\\{1,\ldots,n\\}$, there
are some scalars
$\\{c_{\gamma}^{\prime}\\}_{\gamma\in{\mathbb{F}}_{n}^{+}}\subset{\mathbb{C}}$
with
$\sum_{\gamma\in{\mathbb{F}}_{n}^{+}}|c_{\gamma}^{\prime}|^{2}\frac{1}{b_{\gamma}}<\infty$
such that

$A^{*}e_{\sigma}=\sum_{\gamma\in{\mathbb{F}}_{n}^{+}}c_{\gamma}^{\prime}\frac{\sqrt{b_{\sigma}}}{\sqrt{b_{\sigma\gamma}}}e_{\sigma\gamma}.$

Since
$\left<Ae_{\alpha},e_{\sigma}\right>=\left<e_{\alpha},A^{*}e_{\sigma}\right>$,
the relations above imply

$\left<\sum_{\beta\in{\mathbb{F}}_{n}^{+}}c_{\beta}\frac{\sqrt{b_{\alpha}}}{\sqrt{b_{\alpha\beta}}}e_{\alpha\beta},e_{\sigma}\right>=\left<e_{\alpha},\sum_{\gamma\in{\mathbb{F}}_{n}^{+}}c_{\gamma}^{\prime}\frac{\sqrt{b_{\sigma}}}{\sqrt{b_{\sigma\gamma}}}e_{\sigma\gamma}\right>$

for any $\alpha,\sigma\in{\mathbb{F}}_{n}^{+}$. Assume that
$\sigma=\alpha\beta_{0}$ where $\beta_{0}\in{\mathbb{F}}_{n}^{+}$ and
$|\beta_{0}|\geq 1$. The relation above becomes

$c_{\beta_{0}}=\left<e_{\alpha},\sum_{\gamma\in{\mathbb{F}}_{n}^{+}}c_{\gamma}^{\prime}\frac{\sqrt{b_{\alpha\beta_{0}}}}{\sqrt{b_{\alpha\beta_{0}\gamma}}}e_{\alpha\beta_{0}\gamma}\right>=0.$

Hence $c_{\beta_{0}}=0$ for any $\beta_{0}\in{\mathbb{F}}_{n}^{+}$ with
$|\beta_{0}|\geq 1$. Therefore, $Ae_{\alpha}=c_{g_{0}}e_{\alpha}$ for any
$\alpha\in{\mathbb{F}}_{n}^{+}$, thus $A=c_{g_{0}}I$. This shows that
$C^{*}({\bf W})$ is irreducible.

To prove item (ii), note that one implication is trivial. Assume that
$U:F^{2}(H_{n})\otimes{\mathcal{H}}\to
F^{2}(H_{n})\otimes{\mathcal{H}}^{\prime}$ is a unitary operator such that
$U(W_{i}\otimes I_{\mathcal{H}})=(W_{i}\otimes I_{{\mathcal{H}}^{\prime}})U$
for every $i\in\\{1,\ldots,n\\}$. Then, we have $U(W_{i}^{*}\otimes
I_{\mathcal{H}})=(W_{i}^{*}\otimes I_{{\mathcal{H}}^{\prime}})U$ for
$i\in\\{1,\ldots,n\\}$, and due to the fact that the $C^{*}$-algebra
$C^{*}({\bf W})$ is irreducible, we must have $U=I_{F^{2}(H_{n})}\otimes A$,
where $A\in B({\mathcal{H}},{\mathcal{H}}^{\prime})$ is a unitary operator .
Therefore, $\dim{\mathcal{H}}=\dim{\mathcal{H}}^{\prime}$. To prove item
(iii), note that

$W_{i}W_{i}^{*}e_{\alpha}=\begin{cases}\frac{b_{\gamma}}{b_{g_{i}\gamma}}\,e_{g_{i}\gamma}&\text{
if }\alpha=g_{i}\gamma,\ \,\gamma\in{\mathbb{F}}_{n}^{+}\\\ 0&\text{
otherwise,}\end{cases}$

and $W_{i}^{*}W_{i}e_{\alpha}=\frac{b_{\alpha}}{b_{g_{i}\alpha}}e_{\alpha}$
for all $\alpha\in{\mathbb{F}}_{n}^{+}$. Hence, we deduce that
$W_{i}^{*}W_{i}-\sum_{j=1}^{n}W_{j}W_{j}^{*}$ is a diagonal operator and

$\left(W_{i}^{*}W_{i}-\sum_{j=1}^{n}W_{j}W_{j}^{*}\right)e_{\alpha}=\left(\frac{b_{\alpha}}{b_{g_{i}\alpha}}-\frac{b_{\gamma}}{b_{g_{p}\gamma}}\right)\,e_{\alpha}$

for any $\alpha\in{\mathbb{F}}_{n}^{+}$ with $\alpha=g_{p}\gamma$ for some
$\gamma\in{\mathbb{F}}_{n}^{+}$ and $p\in\\{1,\ldots,n\\}$. Since

$\lim_{\gamma\to\infty}\left(\frac{b_{g_{p}\gamma}}{b_{g_{i}g_{p}\gamma}}-\frac{b_{\gamma}}{b_{g_{p}\gamma}}\right)=0,\quad\text{for
any}\quad p\in\\{1,\ldots,n\\},$

it is clear that the diagonal operator
$W_{i}^{*}W_{i}-\sum_{j=1}^{n}W_{j}W_{j}^{*}$ is compact. Since the
$C^{*}$-algebra $C^{*}({\bf W})$ is irreducible, we conclude that it contains
all the compact operators in $B(F^{2}(H_{n}))$. This completes the proof. ∎

We remark that Theorem 5.4 holds, in particular, for the admissible free
holomorphic functions $g$ such that $F^{2}(g)$ is a normalized unitarily
invariant Hilbert space. In particular, it holds for the free holomorphic
functions provided by Example 4.5 and Example 4.6.

######  Theorem 5.5.

Let $g=1+\sum_{|\alpha|\geq 1}b_{\alpha}Z_{\alpha}$ be a free holomorphic
function in a neighborhood of the origin such that $b_{\alpha}>0$ and
$\sup_{\alpha\in{\mathbb{F}}_{n}^{+}}\frac{b_{\alpha}}{b_{g_{i}\alpha}}<\infty$
for every $i\in\\{1,\ldots,n\\},$ and let ${\bf W}=(W_{1},\ldots,W_{n})$ be
the weighted left creation operators associated with $g$. Assume that the
series
$\sum_{k=0}^{\infty}\sum_{|\alpha|=k}a_{\alpha}W_{\alpha}W_{\alpha}^{*}$ is
convergent in the operator norm topology, where $g^{-1}=1+\sum_{|\alpha|\geq
1}a_{\alpha}Z_{\alpha}$. Then the following statements hold.

  1. (i)

$\boldsymbol{{\mathcal{K}}}(F^{2}(H_{n}))\subset\overline{\text{\rm
span}}\\{W_{\alpha}W_{\beta}^{*}:\ \alpha,\beta\in{\mathbb{F}}_{n}^{+}\\},$
where $\boldsymbol{{\mathcal{K}}}(F^{2}(H_{n}))$ stands for the ideal of all
compact operators in $B(F^{2}(H_{n}))$.

  2. (ii)

There is a unique minimal nontrivial two-sided ideal ${\mathcal{J}}$ of
$C^{*}({\bf W})$. Moreover, ${\mathcal{J}}$ coincides with
$\boldsymbol{{\mathcal{K}}}(F^{2}(H_{n}))$.

  3. (iii)

If, in addition,

$\lim_{\gamma\to\infty}\left(\frac{b_{g_{p}\gamma}}{b_{g_{i}g_{p}\gamma}}-\frac{b_{\gamma}}{b_{g_{p}\gamma}}\right)=0$

for every $i,p\in\\{1,\ldots,n\\}$, then

$C^{*}({\bf W})=\overline{\text{\rm span}}\\{W_{\alpha}W_{\beta}^{*}:\
\alpha,\beta\in{\mathbb{F}}_{n}^{+}\\}.$

###### Proof.

Since $\sum_{k=0}^{\infty}\sum_{|\alpha|=k}a_{\alpha}W_{\alpha}W_{\alpha}^{*}$
is convergent in the operator norm topology, Proposition 1.8 implies

$\sum_{k=0}^{\infty}\sum_{|\alpha|=k}a_{\alpha}W_{\alpha}W_{\alpha}^{*}=P_{{\mathbb{C}}1}\in\overline{\text{\rm
span}}\\{W_{\alpha}W_{\beta}^{*}:\ \alpha,\beta\in{\mathbb{F}}_{n}^{+}\\},$

where $P_{{\mathbb{C}}1}$ is the orthogonal projection of $F^{2}(H_{n})$ onto
${\mathbb{C}}$. For any polynomial $g({\bf W}):=\sum\limits_{|\alpha|\leq
m}d_{\alpha}W_{\alpha}$ and any
$\xi:=\sum\limits_{\beta\in{\mathbb{F}}_{n}^{+}}c_{\beta}e_{\beta}\in
F^{2}(H_{n})$, we have $P_{{\mathbb{C}}1}g({\bf W})^{*}\xi=\left<\xi,g({\bf
W})(1)\right>$ and, consequently,

(5.1) $q({\bf W})P_{{\mathbb{C}}1}g({\bf W})^{*}\xi=\left<\xi,g({\bf
W})(1)\right>q({\bf W})(1)$

for any polynomial $q({\bf W})$. Hence, we deduce that $q({\bf
W})P_{{\mathbb{C}}1}g({\bf W})^{*}$ Is a rank-one operator which belongs to
the operator space $\overline{\text{\rm span}}\\{W_{\alpha}W_{\beta}^{*}:\
\alpha,\beta\in{\mathbb{F}}_{n}^{+}\\}$. Using the definition of the weighted
left creation operators $W_{1},\ldots,W_{n}$, one can see that the set

${\mathcal{E}}:=\left\\{\left(\sum\limits_{|\alpha|\leq
m}d_{\alpha}W_{\alpha}\right)(1):\
m\in{\mathbb{N}},d_{\alpha}\in{\mathbb{C}}\right\\}$

is dense in $F^{2}(H_{n})$, Consequently, relation (5.1) shows that all
compact operators in $B(F^{2}(H_{n}))$ are included in the operator space
$\overline{\text{\rm span}}\\{W_{\alpha}W_{\beta}^{*}:\
\alpha,\beta\in{\mathbb{F}}_{n}^{+}\\}$. This poves part (i).

To prove item (ii), let ${\mathcal{J}}$ be a nontrivial two-sided ideal of
$C^{*}({\bf W})$. Then there exists $A\in{\mathcal{J}}$ such that
$Ae_{\alpha}\neq 0$ for some $\alpha\in{\mathbb{F}}_{n}^{+}$. Note that

$P_{{\mathbb{C}}1}W_{\beta}^{*}e_{\alpha}=\begin{cases}\frac{1}{\sqrt{b_{\beta}}}&\text{
if }\alpha=\beta\\\ 0&\text{ otherwise,}\end{cases}$

implies
$b_{\alpha}W_{\alpha}P_{{\mathbb{C}}1}W_{\alpha}^{*}=P_{{\mathbb{C}}e_{\alpha}}\in\overline{\text{\rm
span}}\\{W_{\alpha}W_{\beta}^{*}:\ \alpha,\beta\in{\mathbb{F}}_{n}^{+}\\}$,
where $P_{{\mathbb{C}}e_{\alpha}}$ is the orthogonal projection of the full
Fock space $F^{2}(H_{n})$ onto the one-dimensional subspace
${\mathbb{C}}e_{\beta}$. This implies that
$P_{{\mathbb{C}}e_{\alpha}}A^{*}AP_{{\mathbb{C}}e_{\alpha}}\in{\mathcal{J}}.$
Note that, for any $x\in F^{2}(H_{n})$, we have

$\begin{split}P_{{\mathbb{C}}e_{\alpha}}A^{*}AP_{{\mathbb{C}}e_{\alpha}}x&=\left<x,P_{{\mathbb{C}}e_{\alpha}}A^{*}Ae_{\alpha}\right>e_{\alpha}=\|Ae_{\alpha}\|^{2}P_{{\mathbb{C}}e_{\alpha}}x.\end{split}$

Since
$\|Ae_{\alpha}\|^{2}P_{{\mathbb{C}}e_{\alpha}}=P_{{\mathbb{C}}e_{\alpha}}A^{*}\Gamma
P_{{\mathbb{C}}e_{\alpha}}$ and $Ae_{\alpha}\neq 0$, we deduce that
$P_{{\mathbb{C}}e_{\alpha}}\in{\mathcal{J}}$. Now, it is clear that
$W_{\alpha}^{*}P_{{\mathbb{C}}e_{\alpha}}W_{\alpha}\in{\mathcal{J}}$ and,
consequently,
$b_{\alpha}W_{\alpha}^{*}P_{{\mathbb{C}}e_{\alpha}}W_{\alpha}=P_{{\mathbb{C}}1}\in{\mathcal{J}}$.
As in the proof of item (i), one can show that $q({\bf
W})P_{{\mathbb{C}}1}g({\bf W})^{*}\in{\mathcal{J}}$ is a rank-one operator for
any polynomials $q({\bf W})$ and $g({\bf W})$. Moreover, this can be used to
show that any rank-one operator in $B(F^{2}(H_{n}))$ is in the ideal
${\mathcal{J}}$. Consequently, we deduce that
$\boldsymbol{{\mathcal{K}}}(F^{2}(H_{n}))\subset{\mathcal{J}}$. Since
${\mathcal{J}}$ is a minimal two-sided ideal of $C^{*}({\bf W})$, we must have
${\mathcal{J}}=\boldsymbol{{\mathcal{K}}}(F^{2}(H_{n}))$.

Now, we prove item (iii). We saw in the proof of Theorem 5.4, part (iii), that
if

$\lim_{\gamma\to\infty}\left(\frac{b_{g_{p}\gamma}}{b_{g_{i}g_{p}\gamma}}-\frac{b_{\gamma}}{b_{g_{p}\gamma}}\right)=0\quad\text{for
every}\quad i,p\in\\{1,\ldots,n\\},$

then $W_{i}^{*}W_{i}-\sum_{j=1}^{n}W_{j}W_{j}^{*}$ is a compact operator in
$B(F^{2}(H_{n}))$. Consequently, using the fact that $W_{i}^{*}W_{j}=0$ if
$i,j\in\\{1,\ldots,n\\}$ with $i\neq j$, we deduce that

$W_{i}^{*}W_{i}W_{i_{1}}\cdots W_{i_{p}}\in\sum_{j=1}^{n}W_{i_{1}}\cdots
W_{i_{p}}W_{j}W_{j}^{*}+\boldsymbol{{\mathcal{K}}}(F^{2}(H_{n}))$

for any $i,i_{1},\ldots,i_{p}$ in $\\{1,\ldots,n\\}$. Inductively, one can
easily prove that

$W_{\beta}^{*}W_{\alpha}\in\text{\rm span}\left\\{W_{\gamma}W_{\sigma}^{*}:\
\gamma,\sigma\in{\mathbb{F}}_{n}^{+}\text{ with
}|\gamma|\leq|\alpha|,|\sigma|\leq|\beta|\right\\}+\boldsymbol{{\mathcal{K}}}(F^{2}(H_{n}))$

for any $\alpha,\beta\in{\mathbb{F}}_{n}^{+}$. On the other hand, according to
the first part of the theorem, we have

$\boldsymbol{{\mathcal{K}}}(F^{2}(H_{n}))\subset\overline{\text{\rm
span}}\left\\{W_{\alpha}W_{\beta}^{*}:\
\alpha,\beta\in{\mathbb{F}}_{n}^{+}\right\\}.$

Combining these results, we deduce that $C^{*}({\bf W})=\overline{\text{\rm
span}}\\{W_{\alpha}W_{\beta}^{*}:\ \alpha,\beta\in{\mathbb{F}}_{n}^{+}\\}.$
This completes the proof. ∎

We remark that Theorem 5.5 holds, in particular, for the admissible free
holomorphic functions of Example 5.3. Here is a $C^{*}$-algebra version of the
Wold decomposition for the unital $*$-representations of the $C^{*}$-algebra
$C^{*}({\bf W})$.

######  Theorem 5.6.

Let $g$ be an admissible free holomorphic function and let ${\bf
W}=(W_{1},\ldots,W_{n})$ be the weighted left creation operators associated
with $g$. If the series
$\sum_{k=0}^{\infty}\sum_{|\alpha|=k}a_{\alpha}W_{\alpha}W_{\alpha}^{*}$ is
convergent in the operator norm topology, where $g^{-1}=1+\sum_{|\alpha|\geq
1}a_{\alpha}Z_{\alpha}$, and $\pi:C^{*}({\bf W})\to B({\mathcal{K}})$ is a
unital $*$-representation of $C^{*}({\bf W})$ on a separable Hilbert space
${\mathcal{K}}$, then $\pi$ decomposes into a direct sum

$\pi=\pi_{0}\oplus\pi_{1}\ \text{ on }\
{\mathcal{K}}={\mathcal{K}}_{0}\oplus{\mathcal{K}}_{1},$

where $\pi_{0}$ and $\pi_{1}$ are disjoint representations of $C^{*}({\bf W})$
on the Hilbert spaces

$\begin{split}{\mathcal{K}}_{0}:&=\overline{\text{\rm
span}}\left\\{\pi(W_{\beta})\Delta_{g^{-1}}(\pi({\bf W}),\pi({\bf
W})^{*}){\mathcal{K}}:\ \beta\in{\mathbb{F}}_{n}^{+}\right\\}\quad\text{ and
}\\\ {\mathcal{K}}_{1}:&={\mathcal{K}}\ominus{\mathcal{K}}_{0},\end{split}$

respectively, where $\pi({\bf W}):=(\pi(W_{1}),\ldots,\pi(W_{n}))$. Moreover,
up to an isomorphism,

${\mathcal{K}}_{0}\simeq
F^{2}(H_{n})\otimes{\mathcal{G}},\quad\pi_{0}(X)=X\otimes
I_{\mathcal{G}}\quad\text{ for }\ X\in C^{*}({\bf W}),$

where ${\mathcal{G}}$ is a Hilbert space with

$\dim{\mathcal{G}}=\dim\left[\text{\rm range}\,\Delta_{g^{-1}}(\pi({\bf
W}),\pi({\bf W})^{*})\right],$

and $\pi_{1}$ is a $*$-representation which annihilates the compact operators
in $C^{*}({\bf W})$ and

$\Delta_{g^{-1}}(\pi_{1}({\bf W}),\pi_{1}({\bf W})^{*})=0.$

If $\pi^{\prime}$ is another $*$-representation of $C^{*}({\bf W})$ on a
separable Hilbert space ${\mathcal{K}}^{\prime}$, then $\pi$ is unitarily
equivalent to $\pi^{\prime}$ if and only if
$\dim{\mathcal{G}}=\dim{\mathcal{G}}^{\prime}$ and $\pi_{1}$ is unitarily
equivalent to $\pi_{1}^{\prime}$.

###### Proof.

According to Theorem 5.5, all the compact operators
$\boldsymbol{{\mathcal{K}}}(F^{2}(H_{n}))$ in $B(F^{2}(H_{n}))$ are contained
in the $C^{*}$-algebra $C^{*}({\bf W})$. Due to standard theory of
representations of $C^{*}$-algebras [5], the representation $\pi$ decomposes
into a direct sum $\pi=\pi_{0}\oplus\pi_{1}$ on
${\mathcal{K}}={\mathcal{K}}_{0}\oplus{\mathcal{K}}_{1}$, where $\pi_{0}$,
$\pi_{1}$ are disjoint representations of $C^{*}({\bf W})$ on the Hilbert
spaces

${\mathcal{K}}_{0}:=\overline{\text{\rm span}}\\{\pi(X){\mathcal{K}}:\
X\in\boldsymbol{{\mathcal{K}}}(F^{2}(H_{n}))\\}\quad\text{ and
}\quad{\mathcal{K}}_{1}:={\mathcal{K}}_{0}^{\perp},$

respectively, such that $\pi_{1}$ annihilates the compact operators in
$B(F^{2}(H_{n}))$ and $\pi_{0}$ is uniquely determined by the action of $\pi$
on the ideal $\boldsymbol{{\mathcal{K}}}(F^{2}(H_{n}))$ of compact operators.
Since every representation of $\boldsymbol{{\mathcal{K}}}(F^{2}(H_{n}))$ is
equivalent to a multiple of the identity representation, we deduce that

${\mathcal{K}}_{0}\simeq{\mathcal{N}}_{J}\otimes{\mathcal{G}},\quad\pi_{0}(X)=X\otimes
I_{\mathcal{G}},\quad X\in C^{*}({\bf W}),$

for some Hilbert space ${\mathcal{G}}$. Using the proof of Theorem 5.5, part
(i), one can see that

$\begin{split}{\mathcal{K}}_{0}&:=\overline{\text{\rm
span}}\\{\pi(X){\mathcal{K}}:\
X\in\boldsymbol{{\mathcal{K}}}(F^{2}(H_{n})))\\}\\\ &=\overline{\text{\rm
span}}\\{\pi(W_{\beta}P_{{\mathbb{C}}1}W_{\alpha}^{*}){\mathcal{K}}:\
\alpha,\beta\in{\mathbb{F}}_{n}^{+}\\}\\\ &=\overline{\text{\rm
span}}\left\\{\pi(W_{\beta})\Delta_{g^{-1}}(\pi({\bf W}),\pi({\bf
W})^{*}){\mathcal{K}}:\ \beta\in{\mathbb{F}}_{n}^{+}\right\\}.\end{split}$

Since $\Delta_{g^{-1}}({\bf W},{\bf W}^{*})=P_{{\mathbb{C}}1}\in C^{*}({\bf
W})$, we have $\Delta_{g^{-1}}(\pi_{1}({\bf W}),\pi_{1}({\bf W})^{*})=0$ and
$\dim{\mathcal{G}}=\dim\left[\text{\rm range}\,\pi(P_{{\mathbb{C}}1})\right].$
To prove the uniqueness, note that according to the standard theory of
representations of $C^{*}$-algebras, $\pi$ and $\pi^{\prime}$ are unitarily
equivalent if and only if $\pi_{0}$ and $\pi_{0}^{\prime}$ (resp. $\pi_{1}$
and $\pi_{1}^{\prime}$) are unitarily equivalent. Using Theorem 5.4, part
(ii), we deduce that $\dim{\mathcal{G}}=\dim{\mathcal{G}}^{\prime}$. The proof
is complete. ∎

Here is our geometric version of the Wold decomposition for unital
$*$-representation of the $C^{*}$-algebra $C^{*}({\bf W})$. This extends the
corresponding result from [47] to our more general setting.

######  Theorem 5.7.

Let $g$ be an admissible free holomorphic function and let ${\bf
W}=(W_{1},\ldots,W_{n})$ be the weighted left creation operators associated
with $g$ such that the series
$\sum_{k=0}^{\infty}\sum_{|\alpha|=k}a_{\alpha}W_{\alpha}W_{\alpha}^{*}$ is
convergent in the operator norm topology, where $g^{-1}=1+\sum_{|\alpha|\geq
1}a_{\alpha}Z_{\alpha}$. Let $\pi$ be a unital $*$-representation of the
$C^{*}$-algebra $C^{*}({\bf W})$ on a separable Hilbert space ${\mathcal{K}}$
and set $V_{i}:=\pi(W_{i})$. Then the noncommutative Berezin kernel
${K_{g,V}}$ is a partial isometry. Setting

${\mathcal{K}}^{(0)}:=\text{\rm range}\,K_{g,V}^{*}\ \text{ and }\
{\mathcal{K}}^{(1)}:=\ker K_{g,V},$

the orthogonal decomposition
${\mathcal{K}}={\mathcal{K}}^{(0)}\bigoplus{\mathcal{K}}^{(1)}$ has the
following properties.

  1. (i)

${\mathcal{K}}^{(0)}$ and ${\mathcal{K}}^{(1)}$ are reducing subspaces for
each operator $V_{i}$.

  2. (ii)

$V|_{{\mathcal{K}}^{(0)}}:=(V_{1}|_{{\mathcal{K}}^{(0)}},\ldots,V_{n}|_{{\mathcal{K}}^{(0)}})$
is a pure tuple in ${\mathcal{D}}_{g^{-1}}({\mathcal{K}}^{(0)})$.

  3. (iii)

$V|_{{\mathcal{K}}^{(1)}}:=(V_{1}|_{{\mathcal{K}}^{(1)}},\ldots,V_{n}|_{{\mathcal{K}}^{(1)}})$
is a Cuntz tuple in ${\mathcal{D}}_{g^{-1}}({\mathcal{K}}^{(1)})$.

Moreover, the Berezin kernel ${K_{g,V}}|_{{\mathcal{K}}^{(0)}}$ is a unitary
operator satisfying relation

$V_{i}|_{{\mathcal{K}}^{(0)}}=\left({K_{g,V}}|_{{\mathcal{K}}^{(0)}}\right)^{*}\left(W_{i}\otimes
I_{{\mathcal{D}}}\right)\left({K_{g,V}}|_{{\mathcal{K}}^{(0)}}\right),\qquad
i\in\\{1,\ldots,n\\},$

where ${\mathcal{D}}:=\text{\rm range}\,\Delta_{g^{-1}}(V,V^{*})$ and
$\Delta_{g^{-1}}(V,V^{*})$ is an orthogonal projection.

In addition, the orthogonal decomposition of ${\mathcal{K}}$ is uniquely
determined by the properties (i), (ii), and (iii) and we have

${\mathcal{K}}^{(0)}=\bigoplus_{\alpha\in{\mathbb{F}}_{n}^{+}}V_{\alpha}({\mathcal{D}}),\qquad\text{where}\quad{\mathcal{D}}={\mathcal{K}}\ominus\left(\bigoplus_{i=1}^{n}\overline{V_{i}{\mathcal{K}}}\right),$

and

${\mathcal{K}}^{(1)}=\bigcap_{s=0}^{\infty}\left(\bigoplus_{\alpha\in{\mathbb{F}}_{n}^{+},|\alpha|=s}\overline{V_{\alpha}({\mathcal{K}})}\right).$

###### Proof.

Due to Theorem 5.6, the Hilbert space ${\mathcal{K}}$ admits an orthogonal
decomposition ${\mathcal{K}}={\mathcal{K}}_{0}\oplus{\mathcal{K}}_{1}$ such
that ${\mathcal{K}}_{0}$ and ${\mathcal{K}}_{1}$ are reducing subspaces for
each operator $V_{i}$ and there is a unitary operator $U:{\mathcal{K}}_{0}\to
F^{2}(H_{n})\otimes{\mathcal{G}}$ with the property that
$V_{i}|_{{\mathcal{K}}_{0}}=U^{*}(W_{i}\otimes I_{\mathcal{G}})U$ for every
$i\in\\{1,\ldots,n\\}$. With respect to the orthogonal decomposition
${\mathcal{K}}={\mathcal{K}}_{0}\oplus{\mathcal{K}}_{1}$, we have
$\Delta_{g^{-1}}(V,V^{*})|_{{\mathcal{K}}_{1}}=0$,

$V_{i}=\left(\begin{matrix}V_{i}|_{{\mathcal{K}}_{0}}&0\\\
0&V_{i}|_{{\mathcal{K}}_{1}}\end{matrix}\right)\quad\text{ and
}\quad\Delta_{g^{-1}}(V,V^{*})=\left(\begin{matrix}\Delta_{g^{-1}}(V,V^{*})|_{{\mathcal{K}}_{0}}&0\\\
0&0\end{matrix}\right).$

Taking into account that the series
$\sum_{k=0}^{\infty}\sum_{|\alpha|=k}a_{\alpha}W_{\alpha}W_{\alpha}^{*}$ is
convergent in the operator norm topology,

$\Delta_{g^{-1}}({\bf W},{\bf
W}^{*})=\sum_{k=0}^{\infty}\sum_{|\alpha|=k}a_{\alpha}W_{\alpha}W_{\alpha}^{*}=P_{{\mathbb{C}}1}$

is an orthogonal projection in $C^{*}({\bf W})$. Since $\pi$ is a
$*$-representation of $C^{*}({\bf W})$, we deduce that the operators
$\Delta_{g^{-1}}(V,V^{*})$ and $\Delta_{g^{-1}}(V,V^{*})|_{{\mathcal{K}}_{0}}$
are orthogonal projections. Consequently, we have

${\mathcal{D}}:=\Delta_{g^{-1}}(V,V^{*}){\mathcal{K}}=\Delta_{g^{-1}}(V,V^{*}){\mathcal{K}}_{0}.$

Since
$b_{\beta}W_{\beta}P_{{\mathbb{C}}1}W_{\beta}^{*}=P_{{\mathbb{C}}e_{\beta}}$
and $\Delta_{g^{-1}}({\bf W},{\bf W}^{*})=P_{{\mathbb{C}}1}\in C^{*}({\bf
W})$, and using the fact that $\pi$ is a $*$-representation of $C^{*}({\bf
W})$, we deduce that the operators
$b_{\beta}^{(m)}V_{\beta}\Delta_{g^{-1}}(V,V^{*})V_{\beta}^{*}$, where
$\beta\in{\mathbb{F}}_{n}^{+}$, are pairwise orthogonal projections and

$K_{g,V}^{*}K_{g,V}=\sum_{\beta\in{\mathbb{F}}_{n}^{+}}b_{\beta}V_{\beta}\Delta_{g^{-1}}(V,V^{*})V_{\beta}^{*}$

is an orthogonal projection. This shows, in particular, that
$V=(V_{1},\ldots,V_{n})\in{\mathcal{D}}_{g^{-1}}({\mathcal{K}})$. On the other
hand, with respect to orthogonal decomposition
${\mathcal{K}}={\mathcal{K}}_{0}\oplus{\mathcal{K}}_{1}$, the noncommutative
Berezin kernel $K_{g,V}$ has the representation

$K_{g,V}=\left[K_{g,V|_{{\mathcal{K}}_{0}}}\
0\right]:{\mathcal{K}}_{0}\oplus{\mathcal{K}}_{1}\to{\mathcal{D}}.$

Since $V|_{{\mathcal{K}}_{0}}$ is a pure $n$-tuple in
${\mathcal{D}}_{g^{-1}}({\mathcal{K}}_{0})$, the Berezin kernel
${K_{g,V|_{{\mathcal{K}}_{0}}}}$ is an isometry satisfying relation

$V_{i}|_{{\mathcal{K}}_{0}}=\left({K_{g,V|_{{\mathcal{K}}_{0}}}}\right)^{*}\left(W_{i}\otimes
I_{{\mathcal{D}}}\right)\left({K_{g,V}}|_{{\mathcal{K}}_{0}}\right),\qquad
i\in\\{1,\ldots,n\\}.$

Moreover, since $V_{i}|_{{\mathcal{K}}_{0}}=U^{*}(W_{i}\otimes
I_{\mathcal{G}})U$ for every $i\in\\{1,\ldots,n\\}$ and $K_{g,{\bf W}\otimes
I_{\mathcal{G}}}$ is a unitary operator, so is
${K_{g,V|_{{\mathcal{K}}_{0}}}}$. Since $K_{g,V}$ is a partial isometry, we
have

${\mathcal{K}}^{(0)}:=\text{\rm range}\,K_{g,V}^{*}={\mathcal{K}}_{0}\ \text{
and }\ {\mathcal{K}}^{(1)}:=\ker K_{g,V}={\mathcal{K}}_{1}.$

The fact that the orthogonal decomposition of ${\mathcal{K}}$ is uniquely
determined by the properties (i), (ii), and (iii) is a consequence of Theorem
5.6. Using the definition of the Berezin kernel $K_{g,V}$, we obtain

${\mathcal{K}}^{(0)}=\left\\{\xi\in{\mathcal{K}}:\
\sum_{\beta\in{\mathbb{F}}_{n}^{+}}b_{\beta}V_{\beta}\Delta_{g^{-1}}(V,V^{*})V_{\beta}^{*}\xi=\xi\right\\}$

and

(5.2) ${\mathcal{K}}^{(1)}=\left\\{\xi\in{\mathcal{K}}:\
\Delta_{g^{-1}}(V,V^{*})V_{\alpha}^{*}\xi=0\ \text{ for every }\
\alpha\in{\mathbb{F}}_{n}^{+}\right\\}.$

Since the operators $b_{\beta}V_{\beta}\Delta_{g^{-1}}(V,V^{*})V_{\beta}^{*}$,
$\beta\in{\mathbb{F}}_{n}^{+}$, are pairwise orthogonal projections, we deduce
that $\sqrt{b_{\beta}}V_{\beta}\Delta_{g^{-1}}(V,V^{*})$ is a partial isometry
with range equal to $V_{\beta}{\mathcal{D}}$ and $V_{\alpha}{\mathcal{D}}\perp
V_{\beta}{\mathcal{D}}$ if $\alpha\neq\beta$. On the other hand, due to
Theorem 5.6, we have ${\mathcal{K}}_{0}=\overline{\text{\rm
span}}\\{V_{\alpha}{\mathcal{D}}:\ \alpha\in{\mathbb{F}}_{n}^{+}\\}$. Now, it
is clear that

(5.3)
${\mathcal{K}}^{(0)}=\bigoplus_{\alpha\in{\mathbb{F}}_{n}^{+}}V_{\alpha}({\mathcal{D}}).$

Due to relation (5.2), if $\xi\in{\mathcal{K}}_{1}$ then
$\Delta_{g^{-1}}(V,V^{*})\xi=0$, which implies
$\xi\in\overline{\operatorname{span}}_{|\alpha|=1}V_{\alpha}{\mathcal{K}}^{(1)}$.
Since ${\mathcal{K}}_{1}$ are reducing subspaces for each operator $V_{i}$, we
obtain
${\mathcal{K}}^{(1)}=\overline{\operatorname{span}}_{|\alpha|=1}V_{\alpha}{\mathcal{K}}^{(1)}$.
Note that the later relation implies
${\mathcal{K}}^{(1)}=\operatorname{span}_{|\alpha|=s}V_{\alpha}{\mathcal{K}}^{(1)}$
for any $s\in{\mathbb{N}}$ and, consequently, we have

(5.4)
${\mathcal{K}}^{(1)}=\bigcap_{s=0}^{\infty}\operatorname{span}_{|\alpha|=s}V_{\alpha}{\mathcal{K}}^{(1)}.$

Using the orthogonal decomposition
${\mathcal{K}}={\mathcal{K}}_{0}\oplus{\mathcal{K}}_{1}$ and that
${\mathcal{K}}_{0}$ and ${\mathcal{K}}_{1}$ are reducing subspaces for each
operator $V_{i}$, one can use the relation (5.4) to prove that

$\bigcap_{s=0}^{\infty}\overline{\operatorname{span}}_{|\alpha|=s}V_{\alpha}{\mathcal{K}}=\left(\bigcap_{s=0}^{\infty}\overline{\operatorname{span}}_{|\alpha|=s}V_{\alpha}{\mathcal{K}}^{(0)}\right)\bigoplus{\mathcal{K}}^{(1)}.$

Due to relation (5.3), we have
$\bigcap_{s=0}^{\infty}\overline{\operatorname{span}}_{|\alpha|=s}V_{\alpha}{\mathcal{K}}^{(0)}=\\{0\\}$.
Therefore,
$\bigcap_{s=0}^{\infty}\overline{\operatorname{span}}_{|\alpha|=s}V_{\alpha}{\mathcal{K}}={\mathcal{K}}^{(1)}.$
Since $V_{i}^{*}V_{j}=0$ for any $i,j\in\\{1,\ldots,n\\}$ with $i\neq j$, we
have
$\overline{\operatorname{span}}_{|\alpha|=s}V_{\alpha}{\mathcal{K}}=\bigoplus_{|\alpha|=s}\overline{V_{\alpha}{\mathcal{K}}}.$
Note that the orthogonal decomposition (5.3) and the fact that
${\mathcal{K}}^{(1)}=\overline{\operatorname{span}}_{|\alpha|=1}V_{\alpha}{\mathcal{K}}^{(1)}$
can be used to show that
${\mathcal{D}}={\mathcal{K}}\ominus\left(\bigoplus_{i=1}^{n}\overline{V_{i}{\mathcal{K}}}\right)$.
This completes the proof. ∎

We remark that Theorem 5.6 holds, in particular, for the admissible free
holomorphic functions provided by Example 5.3.

######  Remark 5.8.

The subspaces ${\mathcal{K}}^{(0)}$ and ${\mathcal{K}}^{(1)}$ in Theorem 5.7
satisfy the relations

${\mathcal{K}}^{(0)}=\left\\{\xi\in{\mathcal{K}}:\
\sum_{\beta\in{\mathbb{F}}_{n}^{+}}b_{\beta}V_{\beta}\Delta_{g^{-1}}(V,V^{*})V_{\beta}^{*}\xi=\xi\right\\}$

and

${\mathcal{K}}^{(1)}=\left\\{\xi\in{\mathcal{K}}:\
\Delta_{g^{-1}}(V,V^{*})V_{\alpha}^{*}\xi=0\ \text{ for every }\
\alpha\in{\mathbb{F}}_{n}^{+}\right\\}.$

Let $\pi$ be a $*$-representation of $C^{*}({\bf W})$ on a Hilbert space
${\mathcal{K}}$ and set $V:=(V_{1},\ldots,V_{n})$, where $V_{i}:=\pi(W_{i})$
for $i\in\\{1,\ldots,n\\}$. We saw in the proof of Theorem 5.7 that if the
series
$\sum_{k=0}^{\infty}\sum_{|\alpha|=k}a_{\alpha}W_{\alpha}W_{\alpha}^{*}$ is
convergent in the operator norm topology, then $\Delta_{g^{-1}}(V,V^{*})$ is
an orthogonal projection and the operators
$b_{\beta}V_{\beta}\Delta_{g^{-1}}(V,V^{*})V_{\beta}^{*}$,
$\beta\in{\mathbb{F}}_{n}^{+}$, are pairwise orthogonal projections. Hence,
$\sum_{\beta\in{\mathbb{F}}_{n}^{+}}b_{\beta}V_{\beta}\Delta_{g^{-1}}(V,V^{*})V_{\beta}^{*}$
is an orthogonal projection and $V\in{\mathcal{D}}_{g^{-1}}({\mathcal{K}})$.

A unital $*$-representation $\pi$ of the $C^{*}$-algebra $C^{*}({\bf W})$ such
that $\Delta_{g^{-1}}(\pi({\bf W}),\pi({\bf W}^{*}))=0,$ is called Cuntz type
representation.

######  Definition 5.9.

Let $g$ be an admissible free holomorphic function and let ${\bf
W}=(W_{1},\ldots,W_{n})$ be the weighted left creation operators associated
with $g$ such that the series
$\sum_{k=0}^{\infty}\sum_{|\alpha|=k}a_{\alpha}W_{\alpha}W_{\alpha}^{*}$ is
convergent in the operator norm topology, where $g^{-1}=1+\sum_{|\alpha|\geq
1}a_{\alpha}Z_{\alpha}$. The algebra ${\mathcal{O}}(g)$ is the universal
$C^{*}$-algebra generated by $\pi(W_{1}),\ldots,\pi(W_{n})$ and the identity,
where $\pi$ is a unital $*$-representation of the $C^{*}$-algebra $C^{*}({\bf
W})$ such that $\Delta_{g^{-1}}(\pi({\bf W}),\pi({\bf W}^{*}))=0,$ where
$\pi({\bf W}):=(\pi(W_{1}),\ldots,\pi(W_{n}))$.

Note that if $g^{-1}=1-(Z_{1}+\cdots+Z_{n})$, $n\geq 2$, then
${\mathcal{O}}(g)$ coincides with the Cuntz algebra ${\mathcal{O}}_{n}$ (see
[16]).

######  Lemma 5.10.

Let $\\{\pi_{\omega}\\}_{\omega}$ be the collection of all Cuntz type
$*$-representations of $C^{*}({\bf W})$ such that $C^{*}(\pi_{\omega}({\bf
W}))$ is irreducible. The universal algebra ${\mathcal{O}}(g)$ is
$*$-isomorphic to the $C^{*}$-algebra $C^{*}(\widetilde{\pi}({\bf W}))$, where
$\widetilde{\pi}:=\bigoplus_{\omega}\pi_{\omega}$ is the direct sum
representation of $C^{*}({\bf W})$.

###### Proof.

Note that $\widetilde{\pi}$ is a Cuntz type $*$-representation of $C^{*}({\bf
W})$ and set $\widetilde{V}:=(\widetilde{V}_{1},\ldots,\widetilde{V}_{n})$,
where $\widetilde{V}_{i}:=\widetilde{\pi}(W_{i})$ for $i\in\\{1,\ldots,n\\}$.
We prove that $C^{*}(\widetilde{V})$ is $*$-isomorphic to ${\mathcal{O}}(g)$.
Let $\pi$ be an arbitrary Cuntz type $*$-representation of $C^{*}({\bf W})$
and set $V:=(V_{1},\ldots,V_{n})$, where $V_{i}:=\pi(W_{i})$ for
$i\in\\{1,\ldots,n\\}$. We need to prove that there is a surjective
$*$-homomorphism $\Omega:C^{*}(\widetilde{V})\to C^{*}(V)$ such that
$\Omega(\widetilde{V}_{i})=V_{i}$. Due to the GNS construction, given a
polynomial $q({\bf Z},{\bf Y}\\})$ in noncommutative indeterminates ${\bf
Z}=\\{Z_{1},\ldots,Z_{n}\\}$ and ${\bf Y}=\\{Y_{1},\ldots,Y_{n}\\}$, there is
an irreducible representation $\gamma$ of $C^{*}(V)$ such that

$\|\gamma\left(q(V,V^{*})\right)\|=\|q(V,V^{*})\|.$

Note that $\gamma\circ\pi$ is an irreducible Cuntz type $*$-representation of
$C^{*}({\bf W})$. Consequently,

$\|q((\gamma\circ\pi)({\bf W}),(\gamma\circ\pi)({\bf
W})^{*})\|\leq\|q(\widetilde{V},\widetilde{V}^{*})\|.$

Combining these relations, we obtain
$\|q(V,V^{*})\|\leq\|q(\widetilde{V},\widetilde{V}^{*})\|,$ for all
polynomials $q({\bf Z},{\bf Y}\\})$, which proves that there is a surjective
$*$-homomorphism $\Omega:C^{*}(\widetilde{V})\to C^{*}(V)$ such that
$\Omega(\widetilde{V}_{i})=V_{i}$. The proof is complete. ∎

The following result is an analogue of Coburn’s theorem [14] for the
$C^{*}$-algebra generated by the unilateral shift on the Hardy space $H^{2}$
of the disk ${\mathbb{D}}=\\{z\in{\mathbb{C}}:|z|<1\\}$ and Cuntz’ extension
[16] for the $C^{*}$-algebra generated by the left creation operators on the
full Fock space with $n$ generators.

######  Theorem 5.11.

Let $g$ be an admissible free holomorphic function with
$g^{-1}=1+\sum_{|\alpha|\geq 1}a_{\alpha}Z_{\alpha}$ and let ${\bf
W}=(W_{1},\ldots,W_{n})$ be the weighted left creation operators associated
with $g$. If
$\sum_{k=0}^{\infty}\sum_{|\alpha|=k}a_{\alpha}W_{\alpha}W_{\alpha}^{*}$ is
convergent in the operator norm topology, then the sequence of
$C^{*}$-algebras

$0\to\boldsymbol{{\mathcal{K}}}(F^{2}(H_{n}))\to C^{*}({\bf
W})\to{\mathcal{O}}(g)\to 0$

is exact, where $\boldsymbol{{\mathcal{K}}}(F^{2}(H_{n}))$ stands for the
ideal of all compact operators in $B(F^{2}(H_{n}))$.

###### Proof.

Due to Theorem 5.5, the two-sided ideal ${\mathcal{J}}$ generated by the
orthogonal projection $\Delta_{g^{-1}}({\bf W},{\bf W})$ in the
$C^{*}$-algebra $C^{*}({\bf W})$ coincides with the ideal
$\boldsymbol{{\mathcal{K}}}(F^{2}(H_{n}))$ of all compact operators in
$B(F^{2}(H_{n}))$. According to Lemma 5.10, it is enough to prove that
$C^{*}(\widetilde{V})$ is $*$-isomorphic to the $C^{*}$-algebra $C^{*}({\bf
W})/_{{\mathcal{J}}}$.

Since the Cuntz type $*$-representation $\widetilde{\pi}:C^{*}({\bf W})\to
C^{*}(\widetilde{V})$ has the property that
$\widetilde{\pi}({\mathcal{J}})=0$, it induces a $*$-representation
$\sigma:C^{*}({\bf W})/_{{\mathcal{J}}}\to C^{*}(\widetilde{V})$ such that
$\sigma\left(q({\bf W},{\bf
W}^{*})+{\mathcal{J}}\right)=q(\widetilde{V},\widetilde{V}^{*})$ for every
polynomial $q({\bf Z},{\bf Y})$ in noncommutative indeterminates ${\bf
Z}=\\{Z_{1},\ldots,Z_{n}\\}$ and ${\bf Y}=\\{Y_{1},\ldots,Y_{n}\\}$.
Therefore, $\sigma$ is surjective and

(5.5) $\|q(\widetilde{V},\widetilde{V}^{*}))\|\leq\|q({\bf W},{\bf
W}^{*})+{\mathcal{J}}\|.$

Let $\rho:C^{*}({\bf W})\to C^{*}({\bf W})/_{{\mathcal{J}}}$ be the canonical
quotient map and note that, due to the fact that ${\mathcal{J}}$ is generated
by the orthogonal projection $\Delta_{g^{-1}}({\bf W},{\bf W})$, we have
$\Delta_{g^{-1}}(\rho({\bf W}),\rho({\bf W}^{*}))=0,$ where $\rho({\bf
W}):=(\rho(W_{1}),\ldots,\rho(W_{n}))$. Therefore, $\rho$ is a Cuntz type
$*$-representation of $C^{*}({\bf W})$ and

$\|q(\rho({\bf W}),\rho({\bf
W}^{*}))\|\leq\|q(\widetilde{V},\widetilde{V}^{*})\|.$

Hence and using relation (5.5), we obtain
$\|q(\widetilde{V},\widetilde{V}^{*})\|=\|q({\bf W},{\bf
W}^{*})+{\mathcal{J}}\|.$ Now, the map defined by

$\Omega(q(\widetilde{V},\widetilde{V}^{*})):=q({\bf W},{\bf
W}^{*})+{\mathcal{J}}$

for all noncommutative polynomials $q({\bf Z},{\bf Y})$ is a contractive
$*$-homomorphism on $C^{*}(\widetilde{V})$ which extends by continuity to a
$*$-homomorphism of $C^{*}(\widetilde{V})$ onto $C^{*}({\bf
W})/_{\mathcal{J}}$. It is clear now that $\Omega$ is a $*$-isomorphism of
$C^{*}$-algebras. The proof is complete. ∎

######  Theorem 5.12.

Let $\pi$ be a unital $*$-representation of $C^{*}({\bf W})$ on a separable
Hilbert space ${\mathcal{K}}$ and let $V:=(V_{1},\ldots,V_{n})$, where
$V_{i}:=\pi({W}_{i})$. If $\pi$ is not a Cuntz type $*$-representation, then
the $C^{*}$-algebras $C^{*}({\bf W})$ and $C^{*}({V})$ are $*$-isomorphic.

###### Proof.

According to Theorem 5.7, we have the Wold decomposition
${\mathcal{K}}={\mathcal{K}}^{(0)}\oplus{\mathcal{K}}^{(1)}$ and
$V_{i}=({W}_{i}\otimes I_{\mathcal{D}})\bigoplus V_{i}^{\prime}$,
$i\in\\{1,\ldots,k\\},$ where $V^{\prime}_{i}:=V_{i}|_{{\mathcal{K}}^{(1)}}$
and ${\mathcal{D}}$ is a Hilbert space with $\dim{\mathcal{D}}\geq 1$.
Moreover, ${V}^{\prime}:=(V_{1}^{\prime},\ldots,V_{k}^{\prime})$ is a Cuntz
$k$-tuple in ${\mathcal{D}}_{g^{-1}}({\mathcal{K}}^{(1)})$. As a consequence,
we deduce that

$q(V,V^{*})=\left(q({\bf W},{\bf W}^{*})\otimes
I_{\mathcal{D}}\right)\bigoplus q(V^{\prime},V^{\prime*})$

for any noncommutative polynomial $q({\bf Z},{\bf Y})$. Since
$\pi|_{{\mathcal{K}}^{(1)}}$ is a $*$-representation, we have
$\|q(V^{\prime},V^{\prime*})\|\leq\|q({\bf W},{\bf W}^{*})\|.$ On the other
hand, since ${\mathcal{D}}\neq\\{0\\}$, we have

$\|q(V,V^{*})\|=\max\left\\{\|q({\bf W},{\bf
W}^{*}))\|,\|q(V^{\prime},V^{\prime*}))\|\right\\}=\|q({\bf W},{\bf W})\|.$

Due to the fact that $\pi:C^{*}({\bf W})\to C^{*}({V})$ is surjective, we have
$\|\pi(g)\|=\|g\|$ for all $g\in C^{*}({\bf W})$. Therefore, $\pi$ is a
$*$-isomorphism of $C^{*}$-algebras. The proof is complete. ∎

######  Theorem 5.13.

Let $\pi_{c}:C^{*}({\bf W})\to B({\mathcal{K}})$ be a Cuntz type
$*$-representation of $C^{*}({\bf W})$ on a Hilbert space ${\mathcal{K}}$, set
$C_{i}:=\pi_{c}({W}_{i})$ and $V_{i}:={W}_{i}\oplus C_{i}$ for
$i\in\\{1,\ldots,n\\}$. Then there is a unique minimal nontrivial two-sided
ideal ${{\mathcal{J}}_{\Delta}}$ of $C^{*}({V})$ and

${{\mathcal{J}}_{\Delta}}=\widetilde{\boldsymbol{{\mathcal{K}}}}\cap
C^{*}({V})=\boldsymbol{{\mathcal{K}}}(F^{2}(H_{n}))\oplus 0,$

where $\widetilde{\boldsymbol{{\mathcal{K}}}}$ is the ideal of all compact
operators in $B(F^{2}(H_{n})\oplus{\mathcal{K}})$. Moreover, if
$V:=(V_{1},\ldots,V_{n})$, then the sequence of $C^{*}$-algebras

$0\to{{\mathcal{J}}_{\Delta}}\to C^{*}({V})\to{\mathcal{O}}(g)\to 0$

is exact.

###### Proof.

Set ${\bf C}:=(C_{1},\ldots,C_{n})$, and note that

$\Delta_{g^{-1}}(V,V^{*})=\Delta_{g^{-1}}({\bf W},{\bf
W}^{*})\bigoplus\Delta_{g^{-1}}(C,C^{*})=P_{{\mathbb{C}}1}\oplus 0.$

This shows that $\widetilde{\boldsymbol{{\mathcal{K}}}}\cap C^{*}({V})$ is a
nontrivial ideal in $C^{*}({V})$. Let ${\mathcal{J}}$ be an arbitrary
nontrivial ideal of $C^{*}({V})$ and let $E\oplus F\in{\mathcal{J}}$ with
$E\oplus F\neq 0$. Then there is a sequence $q_{s}\in{\mathbb{C}}\left<{\bf
Z},{\bf Y}\right>$, $s\in{\mathbb{N}}$, of polynomials such that $q_{s}({\bf
W},{\bf W}^{*})\to E$ and $q_{s}(C,C^{*})\to F$ in the operator norm as
$s\to\infty$. Due to the fact that

$\|q_{s}({\bf W},{\bf W}^{*})\bigoplus q_{s}(C,C^{*})\|\leq\|q_{s}({\bf
W},{\bf W}^{*})\|,$

we deduce that $\|E\oplus F\|\leq\|E\|$. Since $E\oplus F\neq 0$, we deduce
that $E\neq 0$. Consequently, there is $\alpha\in{\mathbb{F}}_{n}^{+}$ such
that $Ee_{\alpha}\neq 0$. As in the proof of Theorem 5.5, part (ii), one can
use the fact that ${C}$ is a Cuntz tuple to show that
$\boldsymbol{{\mathcal{K}}}(F^{2}(H_{n}))\oplus 0\subset{\mathcal{J}}$. This
shows that $\boldsymbol{{\mathcal{K}}}(F^{2}(H_{n}))\oplus 0$ is the minimal
nontrivial ideal of $C^{*}({V})$. Thus
${{\mathcal{J}}_{\Delta}}=\boldsymbol{{\mathcal{K}}}(F^{2}(H_{n}))\oplus 0$.
Since $\boldsymbol{{\mathcal{K}}}(F^{2}(H_{n}))\oplus
0\subset\widetilde{\boldsymbol{{\mathcal{K}}}}\cap C^{*}(V)$, it remains to
prove the reverse inclusion. Indeed, if we fix an operator $E\oplus
F\in\widetilde{\boldsymbol{{\mathcal{K}}}}\cap C^{*}(V)\subset
B(F^{2}(H_{n})\oplus{\mathcal{K}})$, then
$E\in\boldsymbol{{\mathcal{K}}}(F^{2}(H_{n}))$ and $F$ is a compact operator
in $C^{*}({C})\subset B({\mathcal{K}})$. Let $q_{s}\in{\mathbb{C}}\left<{\bf
Z},{\bf Y}\right>$, $s\in{\mathbb{N}}$, be a sequence of polynomials such that
$q_{s}({\bf W},{\bf W}^{*})\to E$ and $q_{s}(C,C^{*})\to F$ in the operator
norm as $s\to\infty$. According to the proof of Theorem 5.11, we have

$\|q_{s}(C,C^{*})\|\leq\|q_{s}({\bf W},{\bf
W}^{*})+\boldsymbol{{\mathcal{K}}}(F^{2}(H_{n}))\|$

which implies $\|F\|\leq\|\rho(E)\|$, where $\rho:C^{*}({\bf W})\to C^{*}({\bf
W})/_{\boldsymbol{{\mathcal{K}}}(F^{2}(H_{n}))}$ is the quotient map. Since
$E$ is a compact operator, we have $\|\rho(E)\|=0$, which implies $F=0$.
Consequently, we have

$\boldsymbol{{\mathcal{K}}}(F^{2}(H_{n}))\oplus
0=\widetilde{\boldsymbol{{\mathcal{K}}}}\cap C^{*}(V).$

The proof of the exactness of the sequence is similar to that of Theorem 5.11.
We leave it to the reader. ∎

Due to the Wold decomposition of Theorem 5.6, the result of Theorem 5.13
carries over to $C^{*}$-algebras $C^{*}(\pi({\bf W}))$, where $\pi:C^{*}({\bf
W})\to B({\mathcal{K}})$ is an arbitrary unital $*$-representation on a
Hilbert space ${\mathcal{K}}$ which is not a Cuntz type $*$-representation. We
remark that Theorem 5.11 and Theorem 5.13 hold, in particular, for the
admissible free holomorphic function provided by Example 5.3.

## References

  * [1] J. Agler, The Arveson extension theorem and coanalytic models, Integral Equations Operator Theory 5 (1982), 608-631.
  * [2] J. Agler, Hypercontractions and subnormality, J. Operator Theory 13 (1985), 203–217.
  * [3] A. Arias and G.Popescu, Noncommutative interpolation and Poisson transforms. II, Houston J. Math. 25 (1999), no. 1, 79–98.
  * [4] A. Arias and G.Popescu, Noncommutative interpolation and Poisson transforms, Israel J. Math. 115 (2000), 205–234.
  * [5] W.B. Arveson, An invitation to $C^{*}$-algebras, Graduate Texts in Math., 39. Springer-Verlag, New-York-Heidelberg, 1976.
  * [6] W.B. Arveson, Subalgebras of $C^{*}$-algebras III: Multivariable operator theory, Acta Math. 181 (1998), 159–228.
  * [7] A. Athavale, On the intertwining of joint isometries, J. Operator Theory 23 (1990), 339–350.
  * [8] J. A. Ball and V. Vinnikov, Lax-Phillips Scattering and Conservative Linear Systems: A Cuntz-Algebra Multidimensional Setting, Mem. Amer. Math. Soc. 837 (2005).
  * [9] J. A. Ball, V. Bolotnikov, A Beurling type theorem in weighted Bergman spaces, C. R. Math. 351 (2013), 433–436.
  * [10] T. Bhattacharyya, J. Eschmeier, J. Sarkar, Characteristic function of a pure commuting contractive tuple, Integral Equations Operator Theory 53 (2005) 23–32.
  * [11] K. Bickel, M. Hartz, and J.E. McCarthy, A multiplier algebra functional calculus, Trans. Amer. Math. Soc. 370 (2018), no. 12, 8467–8482.
  * [12] J. W. Bunce, Models for n-tuples of noncommuting operators, J. Funct. Anal. 57(1984), 21–30.
  * [13] R. Clouâtre and M. Hartz, Multiplier algebras of complete Nevanlinna-Pick spaces: dilations, boundary representations and hyperrigidity, J. Funct. Anal. 274 (2018), no. 6, 1690–1738.
  * [14] L.A. Coburn, The $C^{*}$-algebra generated by an isometry, Bull. Amer. Math. Soc. 73 (1967), 722–726.
  * [15] J.B. Conway, A course in operator theory, Graduate Studies in Mathematics,21, American Mathematical Society, Providence, RI, 2000. xvi+372 pp.
  * [16] J. Cuntz, Simple $C^{*}$–algebras generated by isometries, Commun. Math. Phys. 57 (1977), 173–185.
  * [17] R.E. Curto, F.H. Vasilescu, Automorphism invariance of the operator-valued Poisson transform, Acta Sci. Math. (Szeged) 57 (1993), 65–78.
  * [18] K.R. Davidson and D. Pitts, Invariant subspaces and hyper-reflexivity for free semigroup algebras, Proc. London Math. Soc. 78 (1999), 401–430.
  * [19] K. R. Davidson and D. Pitts, The algebraic structure of non-commutative analytic Toeplitz algebras, Math. Ann. 311 (1998), 275–303.
  * [20] K.R. Davidson, E. Katsoulis, and D. Pitts, The structure of free semigroup algebras, J. Reine Angew. Math. 533 (2001), 99–125.
  * [21] S.W. Drury, A generalization of von Neumann’s inequality to complex ball, Proc. Amer. Math. Soc. 68 (1978), 300–404.
  * [22] A. E. Frazho, Models for noncommuting operators, J. Funct. Anal. 48 (1982), 1–11.
  * [23] I. Katznelson, An introduction to harmonic analysis, Cambridge University Press, Cambridge, 2004.
  * [24] V. Müller, Models for operators using weighted shifts, J. Operator Theory 20 (1988), 3–20.
  * [25] V. Müller and F.H. Vasilescu, Standard models for some commuting multioperators, Proc. Amer. Math. Soc. 117 (1993), 979–989.
  * [26] A. Olofsson, A characteristic operator function for the class of $n$-hypercontractions, J. Funct. Anal. 236 (2006), no. 2, 517–545.
  * [27] A. Olofsson, An operator-valued Berezin transform and the class of $n$-hypercontractions, Integral Equations Operator Theory 58 (2007), no. 4, 503–549.
  * [28] V.I. Paulsen, Completely Bounded Maps and Dilations, Pitman Research Notes in Mathematics, Vol.146, New York, 1986.
  * [29] G. Pisier, Similarity Problems and Completely Bounded Maps, Springer Lect. Notes Math., Vol. 1618, Springer-Verlag, New York, 1995
  * [30] G. Popescu, Isometric dilations for infinite sequences of noncommuting operators, Trans. Amer. Math. Soc. 316 (1989), 523–536.
  * [31] G. Popescu, Characteristic functions for infinite sequences of noncommuting operators, J. Operator Theory 22 (1989), 51–71.
  * [32] G. Popescu, Von Neumann inequality for $(B(H)^{n})_{1}$, Math. Scand. 68 (1991), 292–304.
  * [33] G. Popescu, Functional calculus for noncommuting operators, Michigan Math. J. 42 (1995), 345–356.
  * [34] G. Popescu, Multi-analytic operators on Fock spaces, Math. Ann. 303 (1995), 31–46.
  * [35] G. Popescu, Poisson transforms on some $C^{*}$-algebras generated by isometries, J. Funct. Anal. 161 (1999), 27–61.
  * [36] G. Popescu, Curvature invariant for Hilbert modules over free semigroup algebras, Adv. Math. 158 (2001), 264–309.
  * [37] G. Popescu, Multivariable Nehari problem and interpolation J. Funct. Anal. 200 (2003), no. 2, 536–581.
  * [38] G. Popescu, Operator theory on noncommutative varieties, Indiana Univ. Math. J. 56 (2006), 389–442.
  * [39] G. Popescu, Entropy and Multivariable Interpolation, Mem. Amer. Math. Soc. 184 (2006) no.868 , vi+83 pp.
  * [40] G. Popescu, Free holomorphic functions on the unit ball of $B({\mathcal{H}})^{n}$, J. Funct. Anal. 241 (2006), 268–333.
  * [41] G. Popescu, Noncommutative Berezin transforms and multivariable operator model theory, J. Funct. Anal.  254 (2008), 1003-1057.
  * [42] G. Popescu, Unitary invariants in multivariable operator theory. Mem. Amer. Math. Soc. 200 (2009), no. 941, vi+91 pp.
  * [43] G. Popescu, Free holomorphic automorphisms of the unit ball of $B(H)^{n}$, J. Reine Angew. Math. 638 (2010), 119–168.
  * [44] G. Popescu, Operator theory on noncommutative domains, Mem. Amer. Math. Soc. 205 (2010), no. 964, vi+124 pp.
  * [45] G. Popescu, Invariant subspaces and operator model theory on noncommutative varieties, Math. Ann. 372 (2018), no. 1-2, 611–650.
  * [46] G. Popescu, Bergman spaces over noncommutative domains and commutant lifting, J. Funct. Anal. 280 (2021), no. 8, 108943.
  * [47] G. Popescu, Wold decompositions for representations of $C^{*}$-algebras associated with noncommutative varieties, J. Operator Theory 87 (2022), no. 1, 41–81.
  * [48] G. Popescu, Noncommutative domains, universal operator models, and operator algebras, II, J. Operator Theory, to appear.
  * [49] S. Pott, Standard models under polynomial positivity conditions, J. Operator Theory 41 (1999), no. 2, 365–389.
  * [50] G. Salomon, O.M. Shalit, and E. Shamovich, Algebras of bounded noncommutative analytic functions on subvarieties of the noncommutative unit ball, Trans. Amer. Math. Soc. 370 (2018), no. 12, 8639–8690.
  * [51] G. Salomon, O.M. Shalit, and E. Shamovich, Algebras of noncommutative functions on subvarieties of the noncommutative ball: the bounded and completely bounded isomorphism problem, J. Funct. Anal. 278 (2020), no. 7, 108427, 54 pp.
  * [52] D. Sarason, Generalized interpolation in $H^{\infty}$, Trans. Amer. Math. Soc. 127 (1967), 179–203.
  * [53] B. Sz.-Nagy, C. Foiaş, H. Bercovici, and L. Kérchy, Harmonic Analysis of Operators on Hilbert Space, Second edition. Revised and enlarged edition. Universitext. Springer, New York, 2010. xiv+474 pp.
  * [54] D. Timotin, Regular dilations and models for multicontractions, Indiana Univ. Math. J. 47 (1998), no. 2, 671–-684.
  * [55] F.H. Vasilescu, An operator-valued Poisson kernel, J. Funct. Anal. 110 (1992), 47–72.
  * [56] J. von Neumann, Eine Spectraltheorie für allgemeine Operatoren eines unitären Raumes, Math. Nachr. 4 (1951), 258–281.
  * [57] H. Wold, A study in the analysis of stationary time series, Stockholm, 19338, 2nd ed. 1954.


# Recovering seldom-used theorems of vector calculus and their application to
problems
of electromagnetism.

Antonio Pérez-Garrido<EMAIL_ADDRESS>Dpto. Física Aplicada, Campus
Muralla del Mar, Universidad Politécnica de Cartagena. 30202 Cartagena, Murcia
(Spain)

###### Abstract

In this paper, we use differential forms to prove a number of theorems of
integral vector calculus that are rarely found in textbooks. Two of them, as
far as the author knows, have not been published before. Some possible
applications to problems in physics are shared including a general approach
for computing net forces and torques on current-carrying loops that yields
insights that are not evident from the standard approach.

##  I Introduction

Integral vector calculus theorems provide powerful tools for solving problems
involving electric and magnetic fields. These theorems, such as Gauss’s
theorem

$\oint_{\partial V}\mathbf{A}\cdot
d\mathbf{s}=\int_{V}\mathbf{\nabla}\cdot\mathbf{A}\,dv,$ (1)

where $\mathbf{A}$ is a vector field and $\partial V$ is the boundary surface
of the volume $V$, and Stokes’s theorem

$\oint_{\partial S}\mathbf{A}\cdot
d\mathbf{r}=\int_{S}(\mathbf{\nabla}\times\mathbf{A})\cdot d\mathbf{s},$ (2)

where $\partial S$ is the bounding curve of the surface $S$, allow us, for
example, to switch between integral and differential version of Maxwell
equations.

Other vector integral theorems are much less known, due, likely, to their
lower relevance. Even so, they can be useful in understanding electromagnetic
fields. Three examples are the integral identities

$\oint_{\partial S}\mathbf{A}\times
d\mathbf{r}=\int_{S}\left[(\mathbf{\nabla}\cdot\mathbf{A})d\mathbf{s}-\mathbf{\nabla}(\mathbf{A}\cdot
d\mathbf{s})\right]=-\int_{S}\left(d\mathbf{s}\times\mathbf{\nabla}\right)\times\mathbf{A},$
(3)

$\oint_{\partial S}fd\mathbf{r}=-\int_{S}\mathbf{\nabla}f\times d\mathbf{s},$
(4)

and

$\oint_{\partial S}\left(\mathbf{C}\cdot
d\mathbf{r}\right)\mathbf{A}=\int_{S}d\mathbf{s}\cdot\left(\mathbf{\nabla}\times\mathbf{C}-\mathbf{C}\times\mathbf{\nabla}\right)\mathbf{A},$
(5)

for any scalar field $f$ and vector fields $\mathbf{A}$ and $\mathbf{C}$.
These theorems are rarely found in textbooks about vector calculus.
Furthermore, using (4) and (5) and the expression for the vector triple
product, as detailed later, we also derive a sort of a corollary:

$\oint_{\partial S}\mathbf{C}\times\left(\mathbf{A}\times
d\mathbf{r}\right)=\int_{S}\mathbf{\nabla}\left(\mathbf{C}\cdot\mathbf{A}\right)\times
d\mathbf{s}+\int_{S}\left[d\mathbf{s}\cdot\left(\mathbf{\nabla}\times\mathbf{C}-\mathbf{C}\times\mathbf{\nabla}\right)\right]\mathbf{A}.$
(6)

In all these identities, the orientation of surfaces and curves is crucial.
This is a standard issue in vector integral calculus thus in subsequent
discussions we will not make further mention about this subject, but it is
implicit that one must be cautious when doing calculations.

This work has two purposes. The first is to show how differential forms can be
used to easily demonstrate the identities in Eqs.(3-6). This is done in Sect.
II, which can be skipped by readers who are not already familiar with
differential forms. The second purpose is to show how these identities can be
applied to problems in electromagnetism to reveal new insights that can be
beneficial for teaching physics to advanced undergraduate students, which is
done in Sect. III.

##  II Demonstration of identities

Differential forms are a powerful framework in theoretical physics as
demonstrated, for instance, in their application to quantum mechanics in a
paper by Hoshinohoshino1978 . Additionally, differential forms demonstrate
their robustness for vector analysis and for teaching and understanding
Maxwell’s equations and the principles of electromagnetism amar1980 ;
schleifer1983 ; fumeron2020 ; warnick2014 . In this section, we employ
differential forms as tools for demonstrating identities (3-6). Readers who
are inspired to learn more about differential forms in order to understand
these demonstrations are urged to consult several excellent introductions to
the subject spivak1965 ; needham2021 ; dray2014 .

In this paper, we restrict to $\mathbb{R}^{3}$. Let $A$ be a 1-form, $dx$,
$dy$, $dz$ and $\/\mathbf{i}$, $\/\mathbf{j}$, $\/\mathbf{k}$ orthonormal
bases for 1-forms and vectors, respectively, so a 1-form $A$ can be written as
a linear combination of 1-form basis

$A=A_{x}dx+A_{y}dy+A_{z}dz=\mathbf{A}\cdot d\mathbf{r},$ (7)

where

$\mathbf{A}=A_{x}\,\/\mathbf{i}+A_{y}\,\/\mathbf{j}+A_{z}\,\/\mathbf{k}$ (8)

and

$d\mathbf{r}=dx\,\/\mathbf{i}+dy\,\/\mathbf{j}+dz\,\/\mathbf{k}.$ (9)

Henceforth, the symbol $\wedge$ is not written between elements of the basis
of 1-forms; we will write, for example, $dxdy$ as a short for $dx\wedge dy$,
thus

$dxdy=-dydx\,\,\,{\rm and}\,\,\,dxdx=dydy=dzdz=0.$ (10)

We are going to use the Stokes’s theorem that, formally, says: let $M$ be an
oriented $n$-dimensional smooth manifold with boundary $\partial M$ and let’s
suppose we have an $(n-1)$-form $\alpha$ defined on $M$, the integral of its
exterior derivative, $d\alpha$, over $M$ is equal to the integral of $\alpha$
over the boundary, $\partial M$, of $M$:

$\int_{\partial M}\alpha=\int_{M}d\alpha.$ (11)

The steps to follow for all demonstrations are very simple: compute the
exterior derivative of the integrands of the left hand side of Eqs. (3-6) and
apply (11), in other words, we are going to show that those integrands are the
potentials for the integrands of the right hand side of their respective
equations.

###  II.1 Identity (3)

We can write $\mathbf{A}\times d\mathbf{r}$ as a differential form,

$\mathbf{A}\times d\mathbf{r}=\left(A_{y}dz-
A_{z}dy\right)\/\mathbf{i}+\left(A_{z}dx-
A_{x}dz\right)\/\mathbf{j}+\left(A_{x}dy-A_{y}dx\right)\/\mathbf{k}.$ (12)

Actually, (12) is a vector-valued 1-form. We only have to calculate the
exterior derivative of (12) to complete the proof.

$\begin{split}d(\mathbf{A}\times d\mathbf{r})=&\left[\left(\frac{\partial
A_{y}}{\partial x}dx+\frac{\partial A_{y}}{\partial
y}dy\right)dz-\left(\frac{\partial A_{z}}{\partial x}dx+\frac{\partial
A_{z}}{\partial z}dz\right)dy\right]\/\mathbf{i}+\\\
&\left[\left(\frac{\partial A_{z}}{\partial y}dy+\frac{\partial
A_{z}}{\partial z}dz\right)dx-\left(\frac{\partial A_{x}}{\partial
x}dx+\frac{\partial A_{x}}{\partial y}dy\right)dz\right]\/\mathbf{j}+\\\
&\left[\left(\frac{\partial A_{x}}{\partial x}dx+\frac{\partial
A_{x}}{\partial z}dz\right)dy-\left(\frac{\partial A_{y}}{\partial
y}dy+\frac{\partial A_{y}}{\partial
z}dz\right)dx\right]\/\mathbf{j}.\end{split}$ (13)

We are using the already mentioned fact that after the exterior derivative,
the product between differential forms is the anti-symmetric exterior product
so

$dxdx=dydy=dzdz=0.$ (14)

Regrouping terms in (13) we have

$\begin{split}d(\mathbf{A}\times d\mathbf{r})=&\left[\left(\frac{\partial
A_{x}}{\partial x}+\frac{\partial A_{y}}{\partial y}+\frac{\partial
A_{z}}{\partial z}\right)dydz-\left(\frac{\partial A_{x}}{\partial
x}dydz+\frac{\partial A_{y}}{\partial x}dzdx+\frac{\partial A_{z}}{\partial
x}dxdy\right)\right]\/\mathbf{i}+\\\ &\left[\left(\frac{\partial
A_{x}}{\partial x}+\frac{\partial A_{y}}{\partial y}+\frac{\partial
A_{z}}{\partial z}\right)dzdx-\left(\frac{\partial A_{x}}{\partial
y}dydz+\frac{\partial A_{y}}{\partial y}dzdx+\frac{\partial A_{z}}{\partial
y}dxdy\right)\right]\/\mathbf{j}+\\\ &\left[\left(\frac{\partial
A_{x}}{\partial x}+\frac{\partial A_{y}}{\partial y}+\frac{\partial
A_{z}}{\partial z}\right)dxdy-\left(\frac{\partial A_{x}}{\partial
z}dydz+\frac{\partial A_{y}}{\partial z}dzdx+\frac{\partial A_{z}}{\partial
z}dxdy\right)\right]\/\mathbf{k},\end{split}$ (15)

which can be written in a more compact form as

$d(\mathbf{A}\times
d\mathbf{r})=(\mathbf{\nabla}\cdot\mathbf{A})d\mathbf{s}-\mathbf{\nabla}(\mathbf{A}\cdot
d\mathbf{s}),$ (16)

where $d\mathbf{s}$ is a vector valued two-form representing the differential
surface element, given by:

$d\mathbf{s}=dydz\,\/\mathbf{i}+dzdx\,\/\mathbf{j}+dxdy\,\/\mathbf{k}.$ (17)

If we use the vector triple product identity, the relation (16) can be also
written as

$d(\mathbf{A}\times
d\mathbf{r})=-\left(d\mathbf{s}\times\mathbf{\nabla}\right)\times\mathbf{A}.$
(18)

Using the generalized Stokes’s theorem (11) we arrive to find

$\oint_{\partial S}\mathbf{A}\times
d\mathbf{r}=\int_{S}d\left(\mathbf{A}\times d\mathbf{r}\right),$ (19)

or

$\oint_{\partial S}\mathbf{A}\times
d\mathbf{r}=-\int_{S}\left(d\mathbf{s}\times\mathbf{\nabla}\right)\times\mathbf{A},$
(20)

i.e., equation (3). This theorem can be also demonstrated applying Stokes’s
theorem (2) to vector field $\mathbf{C}=\mathbf{B}\times\mathbf{A}$ for a
constant vector $\mathbf{B}$ and using a number of vector identities lass1950
.

###  II.2 Identity (4)

This identity is the most widely known of (3-6) and can be found in some
textbooks. Following the same procedure than before, we express $fd\mathbf{r}$
as:

$fd\mathbf{r}=fdx\/\mathbf{i}+fdy\/\mathbf{j}+fdz\/\mathbf{k}.$ (21)

Subsequently, we calculate the exterior derivative of (21):

$\begin{split}d(fd\mathbf{r})=&\left(\frac{\partial f}{\partial
y}dy+\frac{\partial f}{\partial z}dz\right)dx\/\mathbf{i}+\left(\frac{\partial
f}{\partial x}dx+\frac{\partial f}{\partial
z}dz\right)dy\/\mathbf{j}+\left(\frac{\partial f}{\partial x}dx+\frac{\partial
f}{\partial y}dy\right)dz\/\mathbf{k}\\\ =&\left(\frac{\partial f}{\partial
z}dzdx-\frac{\partial f}{\partial
y}dxdy\right)\/\mathbf{i}+\left(\frac{\partial f}{\partial
x}dxdy-\frac{\partial f}{\partial
z}dydz\right)\/\mathbf{j}+\left(\frac{\partial f}{\partial
y}dydz-\frac{\partial f}{\partial x}dzdx\right)\/\mathbf{k}\\\
=&-\mathbf{\nabla}f\times d\mathbf{s},\end{split}$ (22)

where we evaluate the differential area element, again, as in expression (17).
Thus, we have that $d(fd\mathbf{r})=-\mathbf{\nabla}f\times d\mathbf{s}$. This
outcome leads us to Eq. (4) upon applying the generalized Stokes’s theorem
(11).

$\oint_{\partial
S}fd\mathbf{r}=\int_{S}d\left(fd\mathbf{r}\right)\Longrightarrow\oint_{\partial
S}fd\mathbf{r}=-\int_{S}\mathbf{\nabla}f\times d\mathbf{s}.$ (23)

As already stated lass1950 ; hsu1984 , the theorems (2–4) can be written in a
unique and compact expression

$\oint_{\partial
S}d\mathbf{r}*\mathbf{g}=\int_{S}\left(d\mathbf{s}\times\mathbf{\nabla}\right)*\mathbf{g},$
(24)

where the asterisk ($*$) denote dot, cross and ordinary products,
respectively. In the latter case $\mathbf{g}$ is, actually, a scalar $g$.

###  II.3 Identity (5)

As before, we calculate the exterior derivative of $(\mathbf{C}\cdot
d\mathbf{r})\mathbf{A}$ and apply (11). Firstly, we develop the expression as
a vector valued differential form:

$(\mathbf{C}\cdot
d\mathbf{r})\mathbf{A}=\left(C_{x}dx+C_{y}dy+C_{z}dz\right)\left(A_{x}\mathbf{i}+A_{y}\mathbf{j}+A_{z}\mathbf{k}\right).$
(25)

For our purposes it is enough to make the exterior derivative of $x$ component
and extend the result to the other two components,

$\begin{split}d\left[\left(C_{x}dx+C_{y}dy+C_{z}dz\right)A_{x}\right]=&C_{x}\left(\frac{\partial
A_{x}}{\partial y}dydx+\frac{\partial A_{x}}{\partial z}dzdx\right)\\\
&+A_{x}\left(\frac{\partial C_{x}}{\partial y}dydx+\frac{\partial
C_{x}}{\partial z}dzdx\right)\\\ &+C_{y}\left(\frac{\partial A_{x}}{\partial
x}dxdy+\frac{\partial A_{x}}{\partial z}dzdy\right)\\\
&+A_{x}\left(\frac{\partial C_{y}}{\partial x}dxdy+\frac{\partial
C_{y}}{\partial z}dzdy\right)\\\ &+C_{z}\left(\frac{\partial A_{x}}{\partial
x}dxdz+\frac{\partial A_{x}}{\partial y}dydz\right)\\\
&+A_{x}\left(\frac{\partial C_{z}}{\partial x}dxdz+\frac{\partial
C_{z}}{\partial y}dydz\right),\end{split}$ (26)

and, after regrouping terms, the right hand side of (26) reduces to

$\begin{split}&\left[\left(C_{y}\frac{\partial}{\partial
x}-C_{x}\frac{\partial}{\partial
y}\right)dxdy+\left(C_{z}\frac{\partial}{\partial
y}-C_{y}\frac{\partial}{\partial
z}\right)dydz+\left(C_{x}\frac{\partial}{\partial
z}-C_{z}\frac{\partial}{\partial x}\right)dzdx\right.\\\
+&\left.\left(\frac{\partial C_{y}}{\partial x}-\frac{\partial C_{x}}{\partial
y}\right)dxdy+\left(\frac{\partial C_{z}}{\partial y}-\frac{\partial
C_{y}}{\partial z}\right)dydz+\left(\frac{\partial C_{x}}{\partial
z}-\frac{\partial C_{z}}{\partial x}\right)dzdx\right]A_{x}.\end{split}$ (27)

For the other two component we only have to change $A_{x}$ in (27) with the
corresponding component $A_{y}$ or $A_{z}$. Thus, we can check that

$d\left((\mathbf{C}\cdot
d\mathbf{r})\mathbf{A}\right)=\left(d\mathbf{s}\cdot\left(\mathbf{\nabla}\times\mathbf{C}-\mathbf{C}\times\mathbf{\nabla}\right)\right)\mathbf{A},$
(28)

where $d\mathbf{s}$ is given by (17) and
$d\mathbf{s}\cdot\left(\mathbf{C}\times\mathbf{\nabla}\right)$ is the 2-form
scalar operator given by

$d\mathbf{s}\cdot\left(\mathbf{C}\times\mathbf{\nabla}\right)=\left(C_{x}\frac{\partial}{\partial
y}-C_{y}\frac{\partial}{\partial
x}\right)dxdy+\left(C_{y}\frac{\partial}{\partial
z}-C_{z}\frac{\partial}{\partial
y}\right)dydz+\left(C_{z}\frac{\partial}{\partial
x}-C_{x}\frac{\partial}{\partial z}\right)dzdx.$ (29)

Thus, appealing again to Stokes theorem (11),

$\oint_{\partial S}\left(\mathbf{C}\cdot
d\mathbf{r}\right)\mathbf{A}=\int_{S}d\left((\mathbf{C}\cdot
d\mathbf{r})\mathbf{A}\right),$ (30)

and, after substitution, we arrive

$\oint_{\partial S}\left(\mathbf{C}\cdot
d\mathbf{r}\right)\mathbf{A}=\int_{S}\left(d\mathbf{s}\cdot\left(\mathbf{\nabla}\times\mathbf{C}-\mathbf{C}\times\mathbf{\nabla}\right)\right)\mathbf{A}.$
(31)

###  II.4 Corollary (6)

Using the vector triple product expression we develop the integrand in the
left side of (6) as

$\mathbf{C}\times\left(\mathbf{A}\times
d\mathbf{r}\right)=\left(\mathbf{C}\cdot
d\mathbf{r}\right)\mathbf{A}-\left(\mathbf{C}\cdot\mathbf{A}\right)d\mathbf{r}.$
(32)

For the first term in the right hand side of (32) we employ (5) and, for the
second term, (4), thus completing the demonstration of corollary (6). The
author has been unable to find either (5) or (6) in any published document,
paper, or textbook.

##  III Application of theorems to problems in electromagnetism

In this section, we apply the established identities, (3–6), to address
problems in electromagnetism. By applying these identities, we aim to unravel
some aspects of the physics that may remain obscured when employing
conventional approaches.

###  III.1 Calculation of the magnetic force on a closed current

The magnetic force, $d\mathbf{F}$, acting on an infinitesimal segment
$d\mathbf{r}$, given by (9), of a loop carrying current $I$, can be expressed
as:

$d\mathbf{F}=Id\mathbf{r}\times\mathbf{B}.$ (33)

For a macroscopic current, the net force is determined by summing up all the
infinitesimal forces,

$\mathbf{F}=I\oint_{\partial S}d\mathbf{r}\times\mathbf{B},$ (34)

where $\partial S$ is the closed curve defining the current loop, and the
boundary of the surface $S$. Applying the first equality in Eq. (3) yields:

$\mathbf{F}=I\int_{S}\mathbf{\nabla}(\mathbf{B}\cdot d\mathbf{s}),$ (35)

where the term involving $\mathbf{\nabla\cdot B}$ has been omitted because the
magnetic field is divergence free and $d\mathbf{s}$ is given by (17). We can
expand (35) to

$\begin{split}\mathbf{F}=I\int_{S}&\left[\left(\frac{\partial B_{x}}{\partial
x}dydz+\frac{\partial B_{y}}{\partial x}dzdx+\frac{\partial B_{z}}{\partial
x}dxdy\right)\/\mathbf{i}\right.\\\ +&\left(\frac{\partial B_{x}}{\partial
y}dydz+\frac{\partial B_{y}}{\partial y}dzdx+\frac{\partial B_{z}}{\partial
y}dxdy\right)\/\mathbf{j}\\\ +&\left.\left(\frac{\partial B_{x}}{\partial
z}dydz+\frac{\partial B_{y}}{\partial z}dzdx+\frac{\partial B_{z}}{\partial
z}dxdy\right)\/\mathbf{k}\/\right].\end{split}$ (36)

In a uniform magnetic field, all the derivatives of $\mathbf{B}$ are zero, and
therefore the net force, $\mathbf{F}$, over the loop is zero. We also see that
the net force will have a non-zero component along an axis only if a component
of the magnetic field is not uniform in that direction. Let see some
particular cases of application of (36).

Figure 1:  A planar current loop in a nonuniform magnetic field. We choose a
frame of reference with the $z$ axis orthogonal to the loop and with the $xy$
plane coincident with the loop.

####  III.1.1 Planar loop

In this case, we can select a frame of reference with its $z$ axis orthogonal
to the plane of the loop, as depicted in Fig. 1. The expression (36) reduces
to

$\mathbf{F}=I\int_{S}\left(\frac{\partial B_{z}}{\partial
x}\mathbf{i}+\frac{\partial B_{z}}{\partial y}\mathbf{j}+\frac{\partial
B_{z}}{\partial z}\mathbf{k}\right)dxdy,$ (37)

as $dydz=dzdx=0$. Consequently, $B_{x}$ and $B_{y}$ do not affect the net
force acting on the current loop. In the particular case that $B_{z}$ only
depends on $z$,

$\frac{\partial B_{z}}{\partial x}=\frac{\partial B_{z}}{\partial
y}=0,\,\,\,\frac{\partial B_{z}}{\partial z}\neq 0,$ (38)

we find the counter intuitive expression

$\mathbf{F}=I\int_{S}\frac{\partial B_{z}}{\partial z}\mathbf{k}dxdy,$ (39)

indicating the emergence of a net magnetic force parallel to the $z$ axis
generated by non-uniformities in $B_{z}$ along the $z$ axis. This seems to
contradict (34), where variations along $z$ axis are not taken into account
since the integral, in this case, is carried out for a fixed $z$. This
apparent contradiction is clarified when we use the fact that the magnetic
field is divergence-free, thus

$\frac{\partial B_{z}}{\partial z}=-\left(\frac{\partial B_{x}}{\partial
x}+\frac{\partial B_{y}}{\partial y}\right),$ (40)

and (39) is given now by

$\mathbf{F}=-I\int_{S}\left(\frac{\partial B_{x}}{\partial x}+\frac{\partial
B_{y}}{\partial y}\right)\mathbf{k}dxdy.$ (41)

Now consider the case of a planar loop placed in the $xy$ plane, as before,
but with $B_{z}$ depending linearly on coordinates:

$B_{z}=b_{x}x+b_{y}y+b_{z}z.$ (42)

After applying (37) we have a magnetic force given by

$\mathbf{F}=IS\left(b_{x}\mathbf{i}+b_{y}\mathbf{j}+b_{z}\mathbf{k}\right),$
(43)

where $S$ is the area of the surface enclosed by the loop.

Figure 2:  A current loop with an arbitrary shape in a magnetic field along
the $z$ direction. The density of field lines may vary depending on the $x$
and $y$ coordinates.

####  III.1.2 Magnetic field with a fixed direction

If the magnetic field points only along the $z$-direction, see Fig. 2, we
obtain once again the expression (37), which is now valid even for a non
planar current loop. Since $B_{x}=B_{y}=0$, the terms $dydz$ and $dzdx$ are
discarded, although they can have non-zero values in the general case, as the
loop may not be flat. In this particular case we have

$\frac{\partial B_{z}}{\partial z}=0,$ (44)

since the magnetic field is divergence free. We are left with the expression

$\mathbf{F}=I\int_{S}\left(\frac{\partial B_{z}}{\partial
x}\mathbf{i}+\frac{\partial B_{z}}{\partial y}\mathbf{j}\right)dxdy,$ (45)

where we can observe that the magnetic force will be perpendicular to the
magnetic field and in the direction of magnetic field non-uniformities.

In the particular case that the magnetic field has a linear dependence on
coordinates:

$B_{z}=b_{x}x+b_{y}y,$ (46)

the magnetic force has the form

$\mathbf{F}=IS_{\rm net}(b_{x}\mathbf{i}+b_{y}\mathbf{j}),$ (47)

where $S_{\rm net}$ is the projected area of the surface enclosed by the loop
onto the $xy$ plane. In general, $S_{\rm net}$ is the given by:

$S_{\rm net}=\int_{S}\mathbf{k}\cdot d\mathbf{s}.$ (48)

When the enclosed surface is flat, $S_{\rm net}=\mathbf{k}\cdot\mathbf{S}$,
where $\mathbf{S}$ is the vector representing the surface enclosed by the
loop, i.e., a vector orthogonal to the surface and with a magnitude equal to
the area of the surface.

###  III.2 Torque on a current-carrying loop

In textbooks, the torque on a flat rectangular electric current loop under the
influence of a uniform magnetic field is usually calculated and then
generalized so that it can be applied to any loop. This generalization
consists of decomposing the current loop into a sum of infinitesimal current
loops, so that the rectangular loop formula is valid for any of those
infinitesimal loops. Here instead we derive the general form of this
expression for a uniform magnetic field $\mathbf{B}$, regardless of the shape
of the loop.

We start be integrating the torque about the origin experienced by an
infinitesimal segment of the loop. The force over an infinitesimal current is
given by (33) and its torque or moment of magnetic force,
$\mathrm{d}\mathbf{M}$, about the origin of the frame of reference, is

$\mathrm{d}\mathbf{M}=\mathbf{r}\times(I\mathrm{d}\mathbf{r}\times\mathbf{B})=I\mathbf{r}\times(\mathrm{d}\mathbf{r}\times\mathbf{B}),$
(49)

where $\mathbf{r}$ is the position vector of the current segment. Using the
corollary (6) in the case of a uniform magnetic field, the total magnetic
torque on an arbitrary current loop can be written

$\begin{split}\mathbf{M}&=I\oint_{\partial
S}\mathbf{r}\times\left(d\mathbf{r}\times\mathbf{B}\right)=-I\oint_{\partial
S}\mathbf{r}\times\left(\mathbf{B}\times d\mathbf{r}\right)\\\
&=-I\int_{S}\mathbf{\nabla}\left(\mathbf{r}\cdot\mathbf{B}\right)\times
d\mathbf{s}-I\int_{S}\left[d\mathbf{s}\cdot\left(\mathbf{\nabla}\times\mathbf{r}-\mathbf{r}\times\mathbf{\nabla}\right)\right]\mathbf{B}.\end{split}$
(50)

Taking into account that the magnetic field is uniform and that
$\mathbf{\nabla}\times\mathbf{r}=0$, the torque reduces to

$\begin{split}\mathbf{M}&=-I\int_{S}\mathbf{\nabla}\left(\mathbf{r}\cdot\mathbf{B}\right)\times
d\mathbf{s}\\\ &=-I\int_{S}\mathbf{\nabla}(xB_{x}+yB_{y}+zB_{z})\times
d\mathbf{s}\\\
&=I\int_{S}d\mathbf{s}\times\left(B_{x}\/\mathbf{i}+B_{y}\/\mathbf{j}+B_{z}\/\mathbf{k}\right)=I\left(\int_{S}d\mathbf{s}\right)\times\mathbf{B}.\end{split}$
(51)

That, for a planar loop becomes

$\mathbf{M}=I\mathbf{S}\times\mathbf{B},$ (52)

where $\mathbf{S}$ is the vector representing the surface enclosed by the
loop. Equation (52) is usually written as

$\mathbf{M}=\mathbf{m}\times\mathbf{B},$ (53)

where $\mathbf{m}=I\mathbf{S}$ receives the name of magnetic moment.

Figure 3:  A planar loop in a non uniform magnetic field with a fixed
direction. We adopt a reference frame with the $z$-axis aligned parallel to
the magnetic field, and the $xz$ plane perpendicular to the planar loop. In
this configuration, the loop is visualized as a singular line. The angle
$\alpha$ represents the inclination between the magnetic field and the normal
to the loop.

An interesting case occurs when considering a planar loop exposed to a non
uniform magnetic field but with a fixed direction, Fig 3. In this situation,
we can select a frame of reference with the $z$ axis aligned with the magnetic
field direction, and with the $xz$ plane perpendicular to the loop. The angle
between the normal to the loop and the direction of $\mathbf{B}$ is denoted
$\alpha$. Similar to the previous case, the magnetic field is dependent solely
on the variables $x$ and $y$:

$\mathbf{B}=B_{z}(x,y)\mathbf{k}.$ (54)

Using (50) and after performing some algebraic manipulations, we derive the
expression for the magnetic torque about the origin of the reference frame as
follows:

$\begin{split}\mathbf{M}=I\int_{S}&\left\\{z\frac{\partial B_{z}}{\partial
y}\cos(\alpha)\mathbf{i}+\left[B_{z}\sin(\alpha)-z\frac{\partial
B_{z}}{\partial x}\cos(\alpha)\right]\mathbf{j}\right.\\\
&\,+\left.\left[\left(x\frac{\partial B_{z}}{\partial y}-y\frac{\partial
B_{z}}{\partial x}\right)\cos(\alpha)-2z\frac{\partial B_{z}}{\partial
y}\sin(\alpha)\right]\mathbf{k}\right\\}ds,\end{split}$ (55)

where we have $dxdy=\cos(\alpha)ds$ and $dydz=\sin(\alpha)ds$. It’s worth
noting that, in this case, the torque about a point it is no longer
independent of the position of that point.

When the loop is orthogonal to the magnetic field, $\alpha=0$, the torque is
given by

$\mathbf{M}=I\int_{S}\left[z\frac{\partial B_{z}}{\partial
y}\mathbf{i}-z\frac{\partial B_{z}}{\partial
x}\mathbf{j}+\left(x\frac{\partial B_{z}}{\partial y}-y\frac{\partial
B_{z}}{\partial x}\right)\mathbf{k}\right]dxdy.$ (56)

Eq. (56) indicates that $x$ and $y$ components of the torque depend on the
nonuniformities of the magnetic field in the $y$ and $x$ directions,
respectively, while the $z$ component depends on both.

On the other hand, when the loop is parallel to the magnetic field,
$\alpha=90^{\rm o}$, the expression (55), becomes

$\mathbf{M}=I\int_{S}\left(B_{z}\,\mathbf{j}+2z\frac{\partial B_{z}}{\partial
y}\mathbf{k}\right)dydz,$ (57)

demonstrating that the torque now lacks a component in the $x$ direction, and
the $y$ component remains independent of the location chosen as the point
about which the torque is calculated.

Eqs. (56) and (57) provide good qualitative insights into how the shape of the
magnetic field can influence the magnetic torque over the current loop, and
give hints about how these identities can offer a comprehensive framework for
analyzing the behavior of current loops in different situations.

##  IV Conclusions

We have presented and demonstrated several theorems, (3–6), which are not
commonly found in standard textbooks on the subject. Notably, a couple of
these theorems, namely (5) and (6), have not been found by the author in
existing publications. Hence, it is plausible that this paper marks the
initial publication of these particular identities. The practical utility of
these theorems has been demonstrated through their application to problems in
electromagnetism, where it is shown how the utilization of these theorems can
offer a more profound understanding of the physics, shedding light on aspects
that may remain hidden when employing more conventional approaches.

##  V Acknowledgements

I would like to express my gratitude to Antonio Arocas for his meticulous
review of the manuscript and his invaluable assistance in researching prior
publications of these theorems in the literature on the subject. I also
acknowledge support from the grant PID2020-120052GB-I00 financed by
MCIN/AEI/10.13039/501100011033.

## References

  * [1] Yoshiaki Hoshino.  An elementary application of exterior differential forms in quantum mechanics.  American Journal of Physics, 46(11):1148–1150, November 1978.
  * [2] Henri Amar.  Elementary application of differential forms to electromagnetism.  American Journal of Physics, 48(3):252–252, March 1980.
  * [3] Nathan Schleifer.  Differential forms as a basis for vector analysis-with applications to electrodynamics.  American Journal of Physics, 51(12):1139–1145, December 1983.
  * [4] S. Fumeron, B. Berche, and F. Moraes.  Improving student understanding of electrodynamics: The case for differential forms.  American Journal of Physics, 88(12):1083–1093, December 2020.
  * [5] Warnick and Russer.  Differential forms and electromagnetic field theory.  Progress In Electromagnetics Research, 148:83–112, 2014.
  * [6] M. Spivak.  Calculus On Manifolds: A Modern Approach To Classical Theorems Of Advanced Calculus.  Avalon Publishing, 1965.
  * [7] T. Needham.  Visual Differential Geometry and Forms: A Mathematical Drama in Five Acts.  Princeton University Press, 2021.
  * [8] T. Dray.  Differential Forms and the Geometry of General Relativity.  Taylor & Francis, 2014.
  * [9] Lass H.  Vector and Tensor Analysis.  Mcgraw-Hill Book Company, Inc., 1950.
  * [10] H.P. Hsu.  Applied Vector Analysis.  Books for Professionals, Inc., 1984.