On the norm-preservation of squares in real algebra representation

One of the main results of the article Gelfand theory for real Banach algebras, recently published in [Rev R Acad Cienc Exactas Fís Nat Ser A Mat RACSAM 114(4):163, 2020] is Proposition 4.1, which establishes that the norm inequality $$\Vert a^{2}\Vert \le \Vert a^{2}+b^{2}\Vert $$ ‖ a 2 ‖ ≤ ‖ a 2 + b 2 ‖ for $$a,b\in {\mathcal {A}}$$ a , b ∈ A is sufficient for a commutative real Banach algebra $${\mathcal {A}}$$ A with a unit to be isomorphic to the space $${\mathcal {C}}_{{\mathbb {R}}}({\mathcal {K}})$$ C R ( K ) of continuous real-valued functions on a compact Hausdorff space $${\mathcal {K}}$$ K . Moreover, in this proposition is also shown that if the above condition (which involves all the operations of the algebra) holds, then the real-algebra isomorphism given by the Gelfand transform preserves the norm of squares. A very natural question springing from the above-mentioned result is whether an isomorphism of $${\mathcal {A}}$$ A onto $${\mathcal {C}}_{{\mathbb {R}}}({\mathcal {K}})$$ C R ( K ) is always norm-preserving of squares. This note is devoted to providing a negative answer to this problem. To that end, we construct algebra norms on spaces $${\mathcal {C}}_{{\mathbb {R}}}({\mathcal {K}})$$ C R ( K ) which are $$(1+\epsilon )$$ ( 1 + ϵ ) -equivalent to the sup-norm and with the norm of the identity function equal to 1, where the norm of every nonconstant function is different from the standard sup-norm. We also provide examples of two-dimensional normed real algebras $${\mathcal {A}}$$ A where $$\Vert a^2\Vert \le k \Vert a^2+b^2\Vert $$ ‖ a 2 ‖ ≤ k ‖ a 2 + b 2 ‖ for all $$a,b\in {\mathcal {A}}$$ a , b ∈ A , for some $$k>1$$ k > 1 , but the inequality fails for $$k=1$$ k = 1 .

On matrix models and their q-deformations

Motivated by the BPS/CFT correspondence, we explore the similarities be- tween the classical β-deformed Hermitean matrix model and the q-deformed matrix models associated to 3d $$ \mathcal{N} $$ N = 2 supersymmetric gauge theories on D2×qS1 and $$ {S}_b^3 $$ S b 3 by matching parameters of the theories. The novel results that we obtain are the correlators for the models, together with an additional result in the classical case consisting of the W -algebra representation of the generating function. Furthermore, we also obtain surprisingly simple expressions for the expectation values of characters which generalize previously known results.

The Duflo non-commutative Fourier transform for the Lorentz group

For a quantum system whose phase space is the cotangent bundle of a Lie group, like for systems endowed with particular cases of curved geometry, one usually resorts to a description in terms of the irreducible representations of the Lie group, where the role of (non-commutative) phase space variables remains obscure. However, a non-commutative Fourier transform can be defined, intertwining the group and (non-commutative) algebra representation, depending on the specific quantization map. We discuss the construction of the non-commutative Fourier transform and the non-commutative algebra representation, via the Duflo quantization map, for a system whose phase space is the cotangent bundle of the Lorentz group.

A Minimal Representation of the Orthosymplectic Lie Supergroup

We construct a minimal representation of the orthosymplectic Lie supergroup $OSp(p,q|2n)$ for $p+q$ even, generalizing the Schrödinger model of the minimal representation of $O(p,q)$ to the super case. The underlying Lie algebra representation is realized on functions on the minimal orbit inside the Jordan superalgebra associated with $\mathfrak{osp}(p,q|2n)$, so that our construction is in line with the orbit philosophy. Its annihilator is given by a Joseph-like ideal for $\mathfrak{osp}(p,q|2n)$, and therefore the representation is a natural generalization of a minimal representation to the context of Lie superalgebras. We also calculate its Gelfand–Kirillov dimension and construct a nondegenerate sesquilinear form for which the representation is skew-symmetric and which is the analogue of an $L^2$-inner product in the supercase.