If $ R $ is a commutative ring, and $ A $ an azumaya algebra is over $ R $, then the switch (or flip or exchange etc.) automorphism of $ A otimes_R A $, given by $ a otimes b mapsto b otimes a $is internal: it is the conjugation by the so-called Goldman element $ g in (A otimes_R A) ^ times $, This element is an important feature of Azumaya algebras and can be defined as the element that corresponds to the reduced trace of $ A $ under the natural $ R $-Modulidentifikation $ A otimes_R A simeq operatorname {End} _R (A) $ (where the reduced trace is seen as a $ R $– linear map $ A to R subset A $).

Now in the context of $ C ^ * $-Algebras if $ R $ is commutative $ C ^ * $-Algebra (corresponding to some $ C_0 (T) $), my understanding is that the equivalent of Azumaya algebras is over $ R $ are the algebras with a continuous trace $ A $ with spectrum $ T $, For example, $ A $ is locally morita equivalent to $ R $;; it also has a Morita inverse: $ A otimes_T bar {A} $ is Morita equivalent to $ R $, Where $ otimes_T $ is the balanced tensor product relative to the spectrum $ T $,

The theory of continuous tracking $ C ^ * $ Algebras have many similarities to the theory of Azumaya algebras, but I couldn't tell if they share this property:

Is there a "Goldman element" $ g in (A otimes_T A) ^ times $ so that the conjugation through $ g $ is the switch automorphism? If so, can this element be chosen consistently?

Since then, the algebraic construction of the element can be imitated almost, but not exactly $ A otimes_T A $ can be identified as $ C_0 (T) $Module with $ mathcal {K} (Y_A) $ Where

$$ Y_A = {a in A , | , (t mapsto tr (aa ^ *) (t)) in C_0 (T) } $$

is a real Hilbert $ C_0 (T) $-Module. This is similar to $ A otimes_R A simeq operatorname {End} _R (A) $ in the algebraic case, however, there is no obvious "trace" element in $ mathcal {K} (Y_A) $,